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#elementary symbolic logic#
This bibliography covers the basics of symbolic logic - the study of formal reasoning through the manipulation of symbols, a topic that includes propositional logic (involving the logical relationships between atomic statements) and first-order predicate logic (which extends this analysis to statements broken down into atomic subjects, predicates, quantifiers, and variable subjects). The study of symbolic logic provides insight into logical fallacies, mathematical proof, and boolean logic, among other scientific and mathematical topics. Higher-order, non-classical, and informal logics are beyond the scope of this bibliography.
**Prerequisites:**
No prerequisite knowledge is necessary to study symbolic logic. Introductory texts will start from the most basic definitions and truth tables for logical operators, so previous experience with logic is not required.
**Where to Start:**
Readers who wish to learn symbolic logic should obtain an elementary textbook. A good text should begin with the basic ideas of propositional logic - that atomic statements can be either true or false, and that truth tables define the operators "and", "or", "not", "if...then", and "if and only if" that combine these atomic statements to express logical relationships. Truth tables involving combinations of statements can be used to demonstrate the logical equivalence of different statements (e.g. "not A and B" is the same as "not-A or not-B"). The resulting logical rules can be used to build arguments - from a set of atomic and complex statements, a valid proof will show that a conclusion must necessarily follow. Truth table analysis will also identify logical fallacies, arguments that appear to be proper but are logically invalid. A classic example is the following: "if it is raining, then the sidewalk is wet" and "the sidewalk is wet" does not imply "it is raining" - this fallacy is known as *affirming the consequent*. Simple intuition tells us that this form is invalid because the sidewalk could be wet for some other reason, and a truth table will verify that these two statements do not imply the conclusion.
After a survey of propositional logic, readers should continue on to study first-order predicate logic. Predicate logic divides atomic statements into subject and predicate; "Spot is a dog" might be represented by "s" in propositional logic, but might be represented by "Ds" in predicate logic, where "D" is the predicate "____ is a dog" and "s" represents the individual constant "Spot". This allows the representation of statements involving universal and existential quantifiers - non-specific statements that refer to either all individuals or at least one individual, respectively. These quantifiers are particularly important in the statement of mathematical theorems.
Readers should read and study each of these topics, but it is extremely important to work many problems as well. Create your own truth tables, try to find equivalent statements of your own, prove the validity of the basic forms of argument, show that fallacies are invalid, and construct your own arguments. Much like mathematics, the only way to internalize the rules of symbolic logic is to practice using them. Upon completing a study of elementary symbolic logic, readers may wish to go on to study further topics in formal logic like non-classical or higher-order logics, a broader study of logic to include informal logic, or methods of mathematical proof as preparation for a study of formal mathematics.
**Books:**
* [Agler, David. **Symbolic Logic: Syntax, Semantics, and Proof**. Rowman & Littlefield Publishers: 2012, 1st ed.](http://www.amazon.com/Symbolic-Logic-Syntax-Semantics-Proof/dp/1442217421)
* [Bergmann, Merrie; Moor, James; and Nelson, Jack. **The Logic Book**. McGraw-Hill Education: 2013, 6th ed.](http://www.amazon.com/Logic-Book-Merrie-Bergmann/dp/0078038413) *(may be a good choice for self-study)*
* [Carnap, Rudolf. **Introduction to Symbolic Logic and Its Applications**. Dover Publications: 2011, 1st English ed.](http://www.amazon.com/Introduction-Symbolic-Logic-Its-Applications/dp/0486604535) *(a second text on symbolic logic for those who have already learned the basics and want more advanced topics)*
* [Forbes, Graeme. **Modern Logic: A Text in Elementary Symbolic Logic**. Oxford University Press: 1994, 1st ed.](http://www.amazon.com/Modern-Logic-Text-Elementary-Symbolic/dp/0195080297)
* [Gensler, Harry J. **Introduction to Logic**. Routledge: 2010, 2nd ed.](http://www.amazon.com/Introduction-Logic-Harry-J-Gensler/dp/0415996511)
* [Gustason, William and Ulrich, Dolph E. **Elementary Symbolic Logic**. Waveland Pr Inc.: 1989, 2nd Sub ed.](http://www.amazon.com/Elementary-Symbolic-Logic-William-Gustason/dp/088133412X)
* [Klenk, Virginia. **Understanding Symbolic Logic**. Pearson: 2007, 5th ed.](http://www.amazon.com/Understanding-Symbolic-Logic-Virginia-Klenk/dp/0132051524)
* [Langer, Susanne K. **An Introduction to Symbolic Logic, 3rd Edition**. Dover Publications: 1967, 3rd ed.](http://www.amazon.com/Introduction-Symbolic-Logic-3rd/dp/0486601641) *(somewhat dated, especially regarding references to Principia Mathematica, but a classic text nonetheless)*
* [Tarski, Alfred. **Introduction to Logic: and to the Methodology of Deductive Sciences**. Important Books: 2013.](http://www.amazon.com/Introduction-Logic-Methodology-Deductive-Sciences/dp/8087830172)
**Articles:**
**Videos:**
* [teachphilosophy's "Logic & Critical Thinking"](https://www.youtube.com/playlist?list=PLFGHE1xQFhhxVI2LmT2yhT_huBZQqsXUA) *(symbolic logic starts at lecture 16, but the previous videos may also be of interest)*
* [Thorsby's "Introduction to Formal Logic"](https://www.youtube.com/playlist?list=PLS8vfA_ckeuZ9UjAHhA1q-ROZGuE_h21V)
**Other Online Sources:**
* [Agler's symbolic logic handouts](http://www.davidagler.com/teaching/logic.html) *(good accompaniment to his textbook)*
* [Carnegie Mellon University's "Logic & Proofs" course](http://oli.cmu.edu/courses/free-open/logic-proofs-course-details/)
* [Green's LogicTutor](http://www.wwnorton.com/college/phil/logic3/)
* [Hardegree's "Symbolic Logic: A First Course (old edition)" (U.Mass.)](http://courses.umass.edu/phil110-gmh/MAIN/IHome-5.htm)
* [Ikenaga's "Rules of Inference and Logic Proofs" (Millersville)](http://www.millersville.edu/~bikenaga/math-proof/rules-of-inference/rules-of-inference.html)
* [Ketland and Schweizer's "Logic 1 Lecture Notes" (Edinburgh)](https://docs.google.com/viewer?docex=1&url=http://www.philosophy.ed.ac.uk/undergraduate/documents/Propositional_Logic_2008_09.pdf)
* [Klement's "Introduction to Logic" course resources (U.Mass.)](http://courses.umass.edu/logic/#lecturenotes)
* [Lane's "Symbolic Logic" lecture notes (West Georgia)](http://www.westga.edu/~rlane/symbolic/)
* [Peacock's "Basic Symbolic Logic" (Lethbridge)](https://docs.google.com/viewer?docex=1&url=http://classes.uleth.ca/200403/logi2003a/SymbolicLogicText.pdf)
* [Wang's "Symbolic Logic Study Guide" (Juniata College)](http://jcsites.juniata.edu/faculty/wang/logic/guide.htm)
* [Wikipedia - "List of Rules of Inference"](https://en.wikipedia.org/wiki/List_of_rules_of_inference)
* [/r/logic](http://np.reddit.com/r/logic/)
* [/r/philosophy](http://np.reddit.com/r/philosophy)
* [/r/askphilosophy](http://np.reddit.com/r/askphilosophy/)
**Subtopics:**
* First-order logic
* Propositional (Sentential) logic
#How to learn our math#
Hi,
So you want to learn math. Fantastic! Math is a wonderful but grueling subject, which is why here we make sure you can have all the resources at your disposal to make sure you either get that A in your class, make math really easy or make sure you really, really know your math to become a mathematician. But say you're our general audience, you're most likely an an Engineering student. Do you really need to learn about topology or abstract algebra? Nope. So this is how to use our math and our suggested guide. Enjoy!
##Engineering
**MechE/Aero/Astro/ChemE/Civil/CompE**
- [Basic Algebra](https://www.reddit.com/r/bibliographies/comments/ajm8wi/basic_algebra/)
- [Precalculus](https://www.reddit.com/r/bibliographies/comments/ajm97d/precalculus/)
- [Methods of Proof](https://www.reddit.com/r/bibliographies/comments/ajq34w/proof_techniques/) *Yes proof techniques for an engineer, it will help understanding proofs in Lnear algebra and Multivariable Calculus*
- [Single Variable Calculus](https://www.reddit.com/r/bibliographies/comments/ajm86t/singlevariable_calculus/)
- [Multivariable Calculus](https://www.reddit.com/r/bibliographies/comments/ak9let/multivariable_calculus/)
- [Linear Algebra](https://old.reddit.com/r/bibliographies/comments/akgoky/linear_algebra/)
- [Differential Equations](https://old.reddit.com/r/bibliographies/comments/akgt8r/differential_equations/)
- Statistics for Engineering and the Sciences
- [Partial Differential Equations](https://old.reddit.com/r/bibliographies/comments/akgtbv/partial_differential_equations/) *Yes you can learn PDE without Real Analysis*
****
**Nuclear/Electrical/ECE**
- [Basic Algebra](https://www.reddit.com/r/bibliographies/comments/ajm8wi/basic_algebra/)
- [Precalculus](https://www.reddit.com/r/bibliographies/comments/ajm97d/precalculus/)
- [Methods of Proof](https://www.reddit.com/r/bibliographies/comments/ajq34w/proof_techniques/) *Yes proof techniques for an engineer, it will help understanding proofs in Lnear algebra and Multivariable Calculus*
- [Single Variable Calculus](https://www.reddit.com/r/bibliographies/comments/ajm86t/singlevariable_calculus/)
- [Multivariable Calculus](https://www.reddit.com/r/bibliographies/comments/ak9let/multivariable_calculus/)
- [Linear Algebra](https://old.reddit.com/r/bibliographies/comments/akgoky/linear_algebra/)
- [Differential Equations](https://old.reddit.com/r/bibliographies/comments/akgt8r/differential_equations/)
- Statistics for Engineering and the Sciences
- [Partial Differential Equations](https://old.reddit.com/r/bibliographies/comments/akgtbv/partial_differential_equations/) *Yes you can learn PDE without Real Analysis*
- [Complex Analysis](https://old.reddit.com/r/bibliographies/comments/axuhxy/complex_analysis/)
##Sciences
**Physics**
- [Basic Algebra](https://www.reddit.com/r/bibliographies/comments/ajm8wi/basic_algebra/)
- [Precalculus](https://www.reddit.com/r/bibliographies/comments/ajm97d/precalculus/)
- [Methods of Proof](https://www.reddit.com/r/bibliographies/comments/ajq34w/proof_techniques/)
- [Single Variable Calculus](https://www.reddit.com/r/bibliographies/comments/ajm86t/singlevariable_calculus/)
- [Multivariable Calculus](https://www.reddit.com/r/bibliographies/comments/ak9let/multivariable_calculus/)
- [Variational Calculus](https://old.reddit.com/r/bibliographies/comments/akgu7e/variational_calculus/)
- [Linear Algebra](https://old.reddit.com/r/bibliographies/comments/akgoky/linear_algebra/)
- [Differential Equations](https://old.reddit.com/r/bibliographies/comments/akgt8r/differential_equations/)
- [Partial Differential Equations](https://old.reddit.com/r/bibliographies/comments/akgtbv/partial_differential_equations/)
- [Tensor Calculus](https://old.reddit.com/r/bibliographies/comments/e3csw3/tensor_calculus/)
- Statistics for Engineering and the Sciences
- [Set Theory](https://old.reddit.com/r/bibliographies/comments/aljhaw/set_theory/)
- [Real Analysis](https://old.reddit.com/r/bibliographies/comments/axuhu3/real_analysis/)
- [Complex Analysis](https://old.reddit.com/r/bibliographies/comments/axuhxy/complex_analysis/)
- [Topology](https://old.reddit.com/r/bibliographies/comments/akguwi/topology/)
- [Differential Geometry](https://old.reddit.com/r/bibliographies/comments/akguun/differential_geometry/)
**Mathematics**
- [Basic Algebra](https://www.reddit.com/r/bibliographies/comments/ajm8wi/basic_algebra/)
- [Precalculus](https://www.reddit.com/r/bibliographies/comments/ajm97d/precalculus/)
- [Methods of Proof](https://www.reddit.com/r/bibliographies/comments/ajq34w/proof_techniques/)
- [Set Theory](https://old.reddit.com/r/bibliographies/comments/aljhaw/set_theory/)
- [Single Variable Calculus](https://www.reddit.com/r/bibliographies/comments/ajm86t/singlevariable_calculus/)
- [Multivariable Calculus](https://www.reddit.com/r/bibliographies/comments/ak9let/multivariable_calculus/)
- [Linear Algebra](https://old.reddit.com/r/bibliographies/comments/akgoky/linear_algebra/)
- [Differential Equations](https://old.reddit.com/r/bibliographies/comments/akgt8r/differential_equations/)
- Statistics
- [Real Analysis](https://old.reddit.com/r/bibliographies/comments/axuhu3/real_analysis/)
- [Partial Differential Equations](https://old.reddit.com/r/bibliographies/comments/akgtbv/partial_differential_equations/)
- [Complex Analysis](https://old.reddit.com/r/bibliographies/comments/axuhxy/complex_analysis/)
- [Topology](https://old.reddit.com/r/bibliographies/comments/akguwi/topology/)
- [Tensor Calculus](https://old.reddit.com/r/bibliographies/comments/e3csw3/tensor_calculus/)
- [Differential Geometry](https://old.reddit.com/r/bibliographies/comments/akguun/differential_geometry/)
- [Variational Calculus](https://old.reddit.com/r/bibliographies/comments/akgu7e/variational_calculus/)
**Chemistry/Biology**
- [Precalculus](https://www.reddit.com/r/bibliographies/comments/ajm97d/precalculus/)
- [Methods of Proof](https://www.reddit.com/r/bibliographies/comments/ajq34w/proof_techniques/)
- [Single Variable Calculus](https://www.reddit.com/r/bibliographies/comments/ajm86t/singlevariable_calculus/)
- [Multivariable Calculus](https://www.reddit.com/r/bibliographies/comments/ak9let/multivariable_calculus/)
- Statistics for Engineering and the Sciences
- [Linear Algebra](https://old.reddit.com/r/bibliographies/comments/akgoky/linear_algebra/)
- [Differential Equations](https://old.reddit.com/r/bibliographies/comments/akgt8r/differential_equations/)
- [Partial Differential Equations](https://old.reddit.com/r/bibliographies/comments/akgtbv/partial_differential_equations/) *Only if you're into computations*
**Computer Science**
- [Precalculus](https://www.reddit.com/r/bibliographies/comments/ajm97d/precalculus/)
- [Methods of Proof](https://www.reddit.com/r/bibliographies/comments/ajq34w/proof_techniques/)
- [Single Variable Calculus](https://www.reddit.com/r/bibliographies/comments/ajm86t/singlevariable_calculus/)
- Statistics
- Discrete Math
- [Linear Algebra](https://old.reddit.com/r/bibliographies/comments/akgoky/linear_algebra/)
#Basic Algebra#
Basic (or elementary) algebra extends arithmetic by introducing symbols known as variables that do not represent a specific number but any number to be inserted later. The goal of algebra is to manipulate expressions that involve these variables in order to study general relationships. For example, the equation "A = lw" can be used to express that the area of *any* rectangle is equal to its length times its width - replacing "l" and "w" with specific measurements will find the area of an actual rectangle. Using algebra, this equation can be changed into "l = A/w", which tells us that the length of any rectangle is equal to its area divided by its width. The ability to easily manipulate algebraic equations in a variety of ways is essential to studying more complex mathematical techniques.
Basic algebra should be distinguished from "algebra" in general, which is a branch of mathematics that manipulates symbols in the context of more complex structures with different properties than ordinary numbers; this more advanced field is sometimes called modern (or abstract) algebra.
**Prerequisites:**
Readers who wish to study basic algebra must have mastered arithmetic. They should have the basic mathematical facts (one-digit addition, subtraction, multiplication, and division) memorized. If these have not been memorized, readers should practice these math facts using flashcards until they can be recited automatically. The most common reason for difficulty in learning algebra is not having a sufficiently strong foundation in basic arithmetic.
**Where to Start:**
Readers should obtain a mathematically-rigorous introductory textbook appropriate to their current level. Textbooks must be read chapter-by-chapter, and it is extremely important to work as many problems found in this text as possible. Just as you can only achieve fluency in a language by speaking it frequently, you can only achieve proficiency in algebra by using it to solve a large number of problems. It may be helpful to purchase additional textbooks that provide additional problems or alternative explanations; there are also online tutorials and videos that might be helpful.
By the end of a study of basic algebra, it is very important that readers be able to manipulate algebraic expressions and equations with fluency. Readers who go on to study higher math will have to simplify and solve algebraic equations while applying more complex techniques, and unless it is second nature, readers will struggle with the algebra instead of learning the new techniques.
After mastering algebra, the next goal is calculus, which is the mathematical foundation of science and its laws. It is possible to go from algebra directly into calculus, but readers may benefit from studying precalculus first - the idea behind this is to cover important topics in trigonometry and analytic geometry beforehand so that students can focus exclusively on the calculus. Going on to study calculus will enable readers to begin learning the physical sciences and engineering.
**Books:**
* [Blitzer, Robert F. **Introductory Algebra for College Students**. Pearson: 2012, 6th ed.](http://www.amazon.com/Introductory-Algebra-College-Students-6th/dp/0321758951) *(first half of a good introductory textbook, covers the basics - has a [solutions manual](http://www.amazon.com/Student-Solutions-Introductory-Algebra-Students/dp/032175980X) available)*
* [Blitzer, Robert F. **Intermediate Algebra for College Students**. Pearson: 2012, 6th ed.](http://www.amazon.com/Intermediate-Algebra-College-Students-Edition/dp/0321758935) *(second part of an introduction to algebra, has a [solutions manual](http://www.amazon.com/Student-Solutions-Intermediate-Algebra-Students/dp/0321760336) available)*
* [Bullock, Gregory. **Algebra in Words: A Guide of Hints, Strategies and Simple Explanations**. Acute Books: 2014.](http://www.amazon.com/ALGEBRA-WORDS-Strategies-Simple-Explanations-ebook/dp/B00JDA8FBE) *(conceptual explanations for topics, great supplement to rigorous text)*
* [Bullock, Gregory. **Algebra in Words 2: MORE Hints, Strategies and Simple Explanations**. Acute Books: 2014.](http://www.amazon.com/ALGEBRA-WORDS-Strategies-Simple-Explanations-ebook/dp/B00S4JF88S)
* [Gelfand, Israel M. and Shen, Alexander. **Algebra**. Birkhäuser: 2002.](http://www.amazon.com/Algebra-Israel-M-Gelfand/dp/0817636773) *(a highly-recommended introductory text)*
* [Huettenmueller, Rhonda. **Algebra DeMYSTiFieD**. McGraw-Hill Professional: 2010, 2nd ed.](http://www.amazon.com/Algebra-DeMYSTiFieD-Second-Rhonda-Huettenmueller/dp/0071743618) *(a good introductory text)*
* [Huettenmueller, Rhonda. **College Algebra DeMYSTiFieD**. McGraw-Hill Professional: 2013, 2nd ed.](http://www.amazon.com/College-Algebra-DeMYSTiFieD-2nd-Edition/dp/0071815848) *(more advanced introductory text)*
* [Kelley, W. Michael. **The Humongous Book of Algebra Problems**. ALPHA: 2008](http://www.amazon.com/gp/product/1592577229) *(illustrates the important techniques of basic algebra through many series of example problems)*
* [Larson, Ron. **College Algebra**. Brooks Cole: 2013, 9th ed.](http://www.amazon.com/College-Algebra-Ron-Larson/dp/1133963021) *(an introductory text recommended for more advanced readers)*
* [McMullen, Chris. **Algebra Essentials Practice Workbook with Answers**.](http://www.amazon.com/Algebra-Essentials-Practice-Workbook-Answers/dp/1453661387) *(contains basic explanations of key topics and many practice problems to solve)*
* [Rappaport, Josh. **Algebra Survival Guide: A Conversational Guide for the Thoroughly Befuddled**. Singing Turtle Press: 2000, 1st ed.](http://www.amazon.com/Algebra-Survival-Guide-Conversational-Thoroughly/dp/0965911381) *(clear, simple explanation of the basics - there is a [workbook](http://www.amazon.com/Algebra-Survival-Guide-Workbook-Understanding/dp/0965911373) available filled with practice problems)*
* [Selby, Peter H. and Slavin, Steve. **Practical Algebra: A Self-Teaching Guide**. John Wiley & Sons: 1991, 2nd ed.](http://www.amazon.com/Practical-Algebra-Self-Teaching-Guide-Second/dp/0471530123) *(might be useful for those looking for an algebra refresher)*
* [Sterling, Mary Jane. **Algebra I for Dummies**. For Dummies: 2010, 2nd ed.](http://www.amazon.com/Algebra-Dummies-Mary-Jane-Sterling/dp/0470559640) *(has an accompanying [workbook](http://www.amazon.com/Algebra-Dummies-Mary-Jane-Sterling/dp/0470559640))*
* [Sterling, Mary Jane. **Algebra II for Dummies**. For Dummies: 2006, 1st ed.](http://www.amazon.com/Algebra-Dummies-Mary-Jane-Sterling/dp/0471775819) *(has an accompanying [workbook](http://www.amazon.com/Algebra-Workbook-Dummies-Mary-Sterling/dp/1118867033) and [book of practice problems](http://www.amazon.com/Algebra-II-Practice-Problems-Dummies/dp/1118446623))*
**Articles:**
**Videos:**
* [Delaware's "College Algebra" lectures (UM - Kansas City)](https://www.youtube.com/watch?v=1Amt_-uB9QQ)
* [KhanAcademy - Algebra basics](https://www.khanacademy.org/math/algebra-basics)
* [KhanAcademy - Algebra I](https://www.khanacademy.org/math/algebra)
* [KhanAcademy - Algebra II](https://www.khanacademy.org/math/algebra2)
* [Leonard's "Intermediate Algebra" lectures](https://www.youtube.com/playlist?list=PLC292123722B1B450) *(good discussion of some algebra concepts with worked examples)*
* [Ron Cox, "Basic Algebra Part 1 and 2"](https://www.youtube.com/watch?v=_NXBMSPrPSQ)
**Other Online Sources:**
* [algebrafree.com](http://algebrafree.com) *(contains worksheets with practice problems and explanations for basic concepts - videos seem to be missing, however)*
* [artofproblemsolving.com's textbooks](http://www.artofproblemsolving.com/store/list.php#all) *(a set of textbooks designed to supplement the standard primary school math curriculum for students entering math competitions)*
* [Chen and Duong's "Elementary Mathematics" lecture notes (Macquarie University)](https://rutherglen.science.mq.edu.au/wchen/lnemfolder/lnem.html)
* [Coolmath.com guides](http://www.coolmath.com/algebra) *(simple explanations of selected topics with worked examples)*
* [Kubota's College Algebra course notes](http://www.msc.uky.edu/ken/ma109/notes.htm)
* [math.com's algebra practice](http://www.math.com/practice/Algebra.html) *(work on the fundamentals of algebra with some good practice problems)*
* [mathopolis.com - Algebra 1 Math Skills Practice](http://www.mathopolis.com/questions/skills.php?year=A1) *(choose a topic and get practice problems)*
* [mathopolis.com - Algebra 2 Math Skills Practice](http://www.mathopolis.com/questions/skills.php?year=A2) *(choose a topic and get practice problems)*
* [Paul's Online Math Notes: Algebra - Practice Problems](http://tutorial.math.lamar.edu/Problems/Alg/Alg.aspx) *(practice problems with solutions)*
* [Purplemath modules](http://www.purplemath.com/modules/) *(short explanations of algebra topics with problems to solve)*
* [West Texas A&M's VirtualMathLab - College Algebra](http://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra/) *(tutorials on important topics in algebra, graphing and several important functions; includes practice tests with answer keys - see also the [beginning algebra](http://www.wtamu.edu/academic/anns/mps/math/mathlab/beg_algebra/) and [intermediate algebra](http://www.wtamu.edu/academic/anns/mps/math/mathlab/int_algebra/index.htm) pages)*
* [List of interactive math websites at /r/learnmath](http://www.reddit.com/r/learnmath/comments/w3q6g/list_of_interactive_math_websites/)
* [/r/math](http://np.reddit.com/r/math)
* [/r/learnmath](http://np.reddit.com/r/learnmath)
* [/r/mathbooks](http://np.reddit.com/r/mathbooks)
* [/r/askmath](https://np.reddit.com/r/askmath)
**Subtopics:**
#Pre-Calculus#
Precalculus encompasses mathematical knowledge useful to those who have taken high school algebra and are preparing to learn university-level calculus. Although readers who have taken algebra *can* move straight into calculus, it is recommended to learn the important background topics from precalculus ahead of time so that readers can focus exclusively on the concepts of calculus. The topics that should be studied in precalculus course can be grouped into three subjects: algebra, trigonometry, and analytic geometry (which should cover the definition of a function).
**Prerequisites:**
Readers should have a solid grasp of arithmetic before attempting to prepare for calculus. It is very important to have basic arithmetic facts memorized - if you struggle with arithmetic, you'll struggle with algebra. If you struggle with algebra, you'll struggle with calculus and so on until you decide that you are someone who is "not good at math". But there are no people who are inherently bad at math, only those who lack sufficient preparation. If you don't have the arithmetic tables memorized, it is very important to get a deck of flashcards and practice until they're automatic.
Readers should also have a basic familiarity with algebra. You should understand the basic rules of algebra and be able to manipulate equations, but may still need to write down every step in solving an equation. Algebra should be practiced diligently alongside the newer topics in trigonometry and analytic geometry. By the time you finish precalculus, you should be able to do algebra quickly and easily.
**Where to Start:**
Readers should obtain a precalculus textbook and work through each of the important topics chapter-by-chapter, solving as many problems as possible at the end of each section. As these books can be pricy, readers may want to purchase older editions, which will be far less expensive. A good preparation for calculus involves three topics - algebra, trigonometry, and analytic geometry. Standard precalculus texts do not focus on algebra, so if more practice with algebra is needed, it is recommended that you also pick up a supplementary text (see also the [basic algebra bibliography](http://www.reddit.com/r/bibliographies/comments/36ltzk/math_basic_algebra/)). Readers who are new to mathematics may find some textbook explanations difficult - use supplemental videos and online materials to get additional information on topics you find difficult. But as with any mathematical technique, the only way to learn is by solving many problems - be sure to work as many problems as possible from your textbook.
*More than anything else, the key to getting prepared for a college-level calculus is being able to manipulate algebraic expressions with fluency.* Readers who struggle with symbols and equations, won't be focused on learning the underlying concepts in calculus. So it is very important to be *comfortable* with algebra before starting calculus. And the only way to do this is to practice algebra correctly - play with the algebra - until it feels natural. You should be able to look at equations like "2/3 x - 9 = 5" and see how these numbers move from one side to the other to end up with "x = 21". Practice your algebra diligently and you will set yourself up for success in calculus.
Trigonometry is encountered in calculus primarily because of its importance in physics and higher math - it is not essential to the concepts of elementary calculus, but will be encountered in problems and examples. Readers should understand what trigonometric ratios are and be able to explain what sine, cosine, and tangent mean using a right triangle inscribed within a unit circle. It will also be helpful to learn how to simplify trigonometric expressions using the most important identities.
In basic Algebraic Geometry, algebraic equations are studied by graphing them in the Cartesian coordinate system. Readers should at least learn the definition of a function, how to graph a function, how to interpret and work with graphs, and the functions associated with the conic sections (e.g. the parabola). There are a standard set of graphs and functions used in calculus as examples such as parabolas, hyperbolas, and the trigonometric functions, and readers should become familiar enough with these to be able to draw them on a graph from their algebraic form.
**Books:**
* [Axler, Sheldon. **Algebra and Trigonometry**. Wiley: 2011, 1st ed.](http://www.amazon.com/Algebra-Trigonometry-Sheldon-Axler/dp/047047081X/) *(contains worked solutions to problems, recommended for self-study; note that text and Axler's Precalculus cover nearly-identical content with slightly different focus)*
* [Axler, Sheldon. **Precalculus: A Prelude to Calculus**. Wiley: 2012, 2nd ed.](http://www.amazon.com/Precalculus-Prelude-Calculus-Sheldon-Axler/dp/047064804X/) *(contains worked solutions to problems, recommended for self-study)*
* [Kelley, W. Michael. **The Humongous Book of Algebra Problems**. ALPHA: 2008](http://www.amazon.com/gp/product/1592577229) *(illustrates the important techniques of basic algebra through many series of example problems)*
* [Kelley, W. Michael. **The Humongous Book of Trigonometry Problems**. ALPHA: 2008](http://www.amazon.com/gp/product/1592577229) *(explains basic trigonometry through example problems)*
* [Kuang, Yang and Kase, Elleyne. **Pre-Calculus for Dummies**. For Dummies: 2012, 2nd ed.](http://www.amazon.com/Pre-Calculus-For-Dummies-Yang-Kuang/dp/1118168887) *(Not as detailed as a textbook, but covers the essentials - has an accompanying [workbook](http://www.amazon.com/Pre-Calculus-Workbook-Dummies-Yang-Kuang/dp/0470923229) of problems)*
* [Larson, Ron. **Precalculus**. Cengage Learning: 2010, '008 ed.](http://www.amazon.com/Precalculus-Ron-Larson-ebook/dp/B00B6D01UE) *(good, detailed coverage of trigonometry, analytic geometry, and functions with useful examples - possibly the best precalculus text to use)*
* [McKeague, Charles P. and Turner, Mark D. **Trigonometry**. Brooks Cole: 2012, 7th ed.](http://www.amazon.com/Trigonometry-Charles-P-McKeague/dp/1111826854) *(an in-depth textbook covering all the essentials of trigonometry with clear explanations and examples)*
* [Neill, Hugh. **Trigonometry - A Complete Introduction: A Teach Yourself Guide**. McGraw-Hill: 2013, 2nd ed.](http://www.amazon.com/Trigonometry---Complete-Introduction-Yourself-Science/dp/1444191144) *(many students have trouble with trigonometry - this supplementary text may be helpful)*
* [Selby, Peter H. and Slavin, Steve. **Practical Algebra: A Self-Teaching Guide**. John Wiley & Sons: 1991, 2nd ed.](http://www.amazon.com/Practical-Algebra-Self-Teaching-Guide-Second/dp/0471530123) *(might be useful for those looking for an algebra refresher)*
* [Simmons, George F. **Precalculus Mathematics in a Nutshell: Geometry, Algebra, Trigonometry**. Wipf & Stock Publishers: 2003.](http://www.amazon.com/Precalculus-Mathematics-Nutshell-Geometry-Trigonometry/dp/1592441300) *(might be useful as a refresher course in precalculus mathematics, but not for those who have never studied these concepts)*
* [Sterling, Mary Jane. **Pre-Calculus for Dummies**. For Dummies: 2014, 1st ed.](http://www.amazon.com/Pre-Calculus-Practice-Problems-Dummies-Online/dp/1118853326) *(1,001 extra problems covering the important topics, with solutions)*
* [Stewart, James; Redlin, Lothar; and Watson, Saleem. **Precalculus: Mathematics for Calculus.** Brooks Cole: 2011, 6th ed.](http://www.amazon.com/Precalculus-Mathematics-Calculus-James-Stewart/dp/0840068077) *(good coverage of topics, but explanations are a bit terse for the intended audience and might go into too much detail in some areas - if you use Stewart, work lots of problems but don't get bogged down)*
**Articles:**
**Videos:**
* [Delaware's "College Algebra" lectures (UM - Kansas City)](https://www.youtube.com/watch?v=1Amt_-uB9QQ)
* [KhanAcademy - Algebra basics](https://www.khanacademy.org/math/algebra-basics)
* [KhanAcademy - Algebra I](https://www.khanacademy.org/math/algebra)
* [KhanAcademy - Algebra II](https://www.khanacademy.org/math/algebra2)
* [KhanAcademy - Trigonometry](https://www.khanacademy.org/math/trigonometry)
* [KhanAcademy - Precalculus](https://www.khanacademy.org/math/precalculus) *(these topics are actually more advanced than needed for elementary calculus, but interesting nonetheless and they will be useful after calculus)*
* [Leonard's "Intermediate Algebra" lectures](https://www.youtube.com/playlist?list=PLC292123722B1B450) *(good discussion of some algebra concepts with worked examples)*
* [midnighttutor's "Trigonometry: The Essentials that You Need for Calculus"](https://www.youtube.com/watch?v=23UX1CM6Q1M) *(comments)*
* [Ron Cox, "Basic Algebra Part 1 and 2"](https://www.youtube.com/watch?v=_NXBMSPrPSQ)
**Other Online Sources:**
* [Chen and Duong's "Elementary Mathematics" lecture notes (Macquarie University)](https://rutherglen.science.mq.edu.au/wchen/lnemfolder/lnem.html)
* [coolmath.com's pre-calculus pages](http://www.coolmath.com/precalculus-review-calculus-intro)
* [Joyce's "Dave's Short Trig Course" (Clark University)](http://www.clarku.edu/~djoyce/trig/)
* [Lamar University's "Trig Cheat Sheet"](https://docs.google.com/viewer?docex=1&url=http://tutorial.math.lamar.edu/pdf/Trig_Cheat_Sheet_Reduced.pdf)
* [math.com's algebra practice](http://www.math.com/practice/Algebra.html) *(brush up on the fundamentals of algebra with some good practice problems)*
* [mathisfun.com's "Introduction to Trigonometry" page](https://www.mathsisfun.com/algebra/trigonometry.html)
* [mathisfun.com's "Random Trigonometry Problems"](https://www.mathsisfun.com/games/random-trigonometry.html) *(generates trigonometry problems to practice)*
* [Mueller's "Exploring Precalculus" page](http://wmueller.com/precalculus/) *(has a few good conceptual explanations of functions and rates of change, and the "Am I Ready for Calculus?" page is a good read)*
* [New Planet School's YouTube videos on trigonometry](https://www.youtube.com/playlist?list=PLmSGbjacooPfwaA5s3NYox9AIrXFwFsNW)
* [Paul's Online Math Notes: Algebra - Practice Problems](http://tutorial.math.lamar.edu/Problems/Alg/Alg.aspx) *(practice problems with solutions)*
* [Purplemath modules](http://www.purplemath.com/modules/) *(short explanations of algebra and trigonometry topics with problems to solve)*
* [themathpage.com's "Topics in Precalculus"](http://www.themathpage.com/aprecalc/precalculus.htm) *(focuses on functions, has somewhat terse explanations but includes review questions)*
* [West Texas A&M's VirtualMathLab - College Algebra](http://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra/) *(tutorials on important topics in algebra, graphing and several important functions; includes practice tests with answer keys - see also the [beginning algebra](http://www.wtamu.edu/academic/anns/mps/math/mathlab/beg_algebra/) and [intermediate algebra](http://www.wtamu.edu/academic/anns/mps/math/mathlab/int_algebra/index.htm) pages)*
* [List of interactive math websites at /r/learnmath](http://www.reddit.com/r/learnmath/comments/w3q6g/list_of_interactive_math_websites/)
* [/r/math](http://np.reddit.com/r/math)
* [/r/learnmath](http://np.reddit.com/r/learnmath)
* [/r/mathbooks](http://np.reddit.com/r/mathbooks)
* [/r/askmath](https://np.reddit.com/r/askmath)
**Subtopics:**
#Proof Techniques#
Proof is essential to the structure of mathematics; it provides mathematical statements with a certainty that is impossible in virtually every other field of intellectual inquiry. A valid proof provides an absolute link between established axioms and truths of mathematics and a new piece of mathematical knowledge known as a theorem. Proficiency with these techniques is a prerequisite to the study of higher mathematics. This bibliography covers the basic methods that are used to contruct a proof of a theorem, while proof theory, computer-assisted proof, and other topics in mathematical logic are outside its scope.
**Prerequisites:**
Readers can study methods of proof without any prior knowledge. However, familiarity with basic propositional and first order logic may be helpful, since proofs are essentially informal arguments with an underlying formal logical structure. For example, one of the basic proof techniques is proving the contrapositive rather than the original statement of a theorem, and readers who have studied logic will immediately understand why the contrapositive is logically equivalent to the conditional statement itself. Many introductory proof textbooks will contain these aspects of formal logic, so a separate study is not strictly necessary.
It is difficult to demonstrate the methods of proof without having something to prove, and so different introductory texts will typically assume (or explain) some background mathematical knowledge. Readers should check that the sources they use do not assume too much knowledge beyond their current level; however this will not usually pose an insurmountable problem for those familiar with elementary mathematics and algebra.
**Where to Start:**
Readers wanting to learn how to construct proofs should obtain an introductory textbook. Proof techniques should be learned in two steps: first understand how the strategy works, then use that technique to prove simple mathematical statements until the proof strategy becomes second nature. For example, to understand proof by contradiction you must first understand the idea behind the technique - statements can only be true or false, so if you can demonstrate that it is impossible for a statement to be false by deriving a contradicton, then the statement must be true - then practice it by proving statements; the classic example of proof by contradiction is the proof that the square root of two is irrational: if you assume that the square root of two is a reduced fraction a/b, you can show that a,b must have the factor 2 in common, which contradicts the assumption that a/b is a reduced fraction, and therefore the square root of two must be irrational. Choose many simple mathematical statements and practice using each strategy several times.
Readers who complete a study of proof methods should understand the conditional structure of theorems, understand how to write concise proofs, and know the following proof methods: direct proof, proof by contradiction, proving the contrapositive, proof by exhaustion (cases), existence and uniqueness proofs, universal and existential quantifiers and counterexamples, proving biconditional statements, and mathematical induction. After completing this study, readers will be prepared to study formal mathematics, although it is advisible to study basic math through elementary calculus before beginning work on pure mathematics. Good places to start are real analysis, discrete mathematics, or number theory.
**Books:**
* [Chartrand, Gary; Polimeni, Albert D.; and Zhang, Ping. **Mathematical Proofs: A Transition to Advanced Mathematics**. Pearson: 2012, 3rd ed.](http://www.amazon.com/Mathematical-Proofs-Transition-Advanced-Mathematics/dp/0321797094) *(many recommendations)*
* [Hammack, Richard. **Book of Proof**. Self-published: 2013, revised edition.](http://www.amazon.com/Book-Proof-Richard-Hammack/dp/0989472108) *(available online [here](http://www.people.vcu.edu/~rhammack/BookOfProof/))*
* [Houston, Kevin. **How to Think Like a Mathematician: A Companion to Undergraduate Mathematics**. Cambridge University Press: 2009, 1st ed.](http://www.amazon.com/How-Think-Like-Mathematician-Undergraduate/dp/052171978X)
* [Polya, G. and Conway, John H. **How to Solve It: A New Aspect of Mathematical Method**. Princeton University Press: 2014, reprint edition.](http://www.amazon.com/How-Solve-It-Mathematical-Princeton/dp/069116407X) *(a classic on mathematical problem solving, but not specifically proof techniques - good as a supplementary text)*
* [Solow, Daniel. **How to Read and Do Proofs: An Introduction to Mathematical Thought Processes**. Wiley: 2013, 6th ed.](http://www.amazon.com/How-Read-Proofs-Introduction-Mathematical/dp/1118164024) *(highly recommended)*
* [Taylor, John and Garnier, Rowan. **Understanding Mathematical Proof**. Chapman and Hall/CRC: 2014, 1st ed.](http://www.amazon.com/Understanding-Mathematical-Proof-John-Taylor/dp/1466514906)
* [Velleman, Daniel J. **How to Prove It**. Cambridge University Press: 2006, 2nd ed.](http://www.amazon.com/How-Prove-Structured-Approach-2nd/dp/0521675995)
* [Wolf, Robert S. **Proof, Logic, and Conjecture: The Mathematician's Toolbox**. W. H. Freeman: 1998.](http://www.amazon.com/Proof-Logic-Conjecture-Mathematicians-Toolbox/dp/0716730502)
**Articles:**
* [Jensen-Vallin, Jacqueline A. "Notes for a Course on Proofs". JIBLM **27** (2012).](http://www.jiblm.org/downloads/dlitem.php?id=88&category=jiblmjournal) *(filled with good, simple mathematical statements to prove)*
* [Taylor, Ron. "Introduction to Proof". JIBLM **4** (2007).](http://www.jiblm.org/downloads/dlitem.php?id=56&category=jiblmjournal) *(a short introduction to proof)*
**Videos:**
* [Shillito's Introduction to Higher Mathematics videos, "Lecture 4: Proof Techniques"](https://www.youtube.com/watch?v=WSb9q4Rj2Bg)
* [Shillito's Introduction to Higher Mathematics videos, "Lecture 7: More Proof Techniques"](https://www.youtube.com/watch?v=0UgID8C9RvE)
**Other Online Sources:**
* [Aboutabl's "Methods of Proof" notes](https://docs.google.com/viewer?docex=1&url=https://users.cs.jmu.edu/aboutams/Web/DiscreteMath/Fall03/Lecture%20Notes/1_5%20Methods%20of%20Proofs.pdf)
* [Binegar's Analysis lecture notes (lectures 1-4) (Oklahoma State)](https://math.okstate.edu/people/binegar/4023/4023-lec.html)
* [California State University - "Notes on Methods of Proof"](http://www.math.csusb.edu/notes/proofs/pfnot/pfnot.html)
* [Chakrabarty's "Proofs by Contradiction and by Mathematical Induction" notes (Dartmouth)](https://docs.google.com/viewer?docex=1&url=http://www.cs.dartmouth.edu/~ac/Teach/CS19-Winter06/SlidesAndNotes/lec12induction.pdf) *(explanation and examples of direct proof, indirect proof, and induction)*
* [Coursera's "Introduction to Mathematical Thinking" (Stanford)](https://www.coursera.org/course/maththink)
* [Cusick's "How to Write Proofs" (Fresno State)](http://zimmer.csufresno.edu/~larryc/proofs/proofs.html)
* [Gallian's "Advice for Students Learning Proofs" (Minnesota-Duluth)](http://www.d.umn.edu/~jgallian/Proofs.html)
* [Hagen's "Tips for Proofs" (Virginia Tech)](http://www.math.vt.edu/people/hagen/Fall/Math4334/Tips.html)
* [Hayes' "Proof by Induction" (UCLA)](https://docs.google.com/viewer?docex=1&url=http://www.math.ucla.edu/~brh6/Induction.pdf) *(good overview of theory behind induction proofs)*
* [Hefferon's "Introduction to Proofs, an Inquiry-Based Approach" (St Michael's College)](http://joshua.smcvt.edu/proofs/) *(learn proofs by working through selected proofs)*
* [Heil's "Writing Proofs" (Georgia Tech)](https://docs.google.com/viewer?docex=1&url=http://people.math.gatech.edu/~heil/handouts/proofs.pdf)
* [Henning's "An Introduction to Logic and Proof Techniques" (KwaZulu-Natal)](https://docs.google.com/viewer?docex=1&url=http://www.maths.unp.ac.za/coursework/math130/Notes/LogicProofNotes.pdf)
* [Hsu's "Writing Proofs" (San Jose State)](https://docs.google.com/viewer?docex=1&url=http://www.math.sjsu.edu/~hsu/courses/generic/proof.pdf)
* [Hutchings' "Introduction to Mathematical Arguments" (Berkeley)](https://docs.google.com/viewer?docex=1&url=https://math.berkeley.edu/~hutching/teach/proofs.pdf)
* [Lee's "Some Remarks on Writing Mathematical Proofs" (University of Washington)](https://docs.google.com/viewer?docex=1&url=http://www.math.washington.edu/~lee/Writing/writing-proofs.pdf)
* [Meyer's "Mathematics for Computer Science" notes, chapter 1 (MIT)](http://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-042j-mathematics-for-computer-science-spring-2010/readings/)
* [Pitman's "Notes on Proof Techniques (Other Than Induction)" (Cortland)](https://docs.google.com/viewer?docex=1&url=http://web.cortland.edu/pitmand/224/notesProof.pdf)
* [Sundstrom's "Mathematical Reasoning: Writing and Proof" (Grand Valley State)](http://scholarworks.gvsu.edu/books/7/) *(an open textbook on proofs, contains a good list of proof guidelines in the appendix)*
* [Wilde's "Math Camp Notes: Basic Proof Techniques" (South Florida)](https://docs.google.com/viewer?docex=1&url=http://faculty.cas.usf.edu/jkwilde/mathcamp/Basic_Proof_Techniques.pdf) *(good summary with a few mathematical statements to practice proofs)*
* [/r/mathriddles](http://np.reddit.com/r/mathriddles) *(a good place to find mathematical statements to prove)*
* [/r/math](http://np.reddit.com/r/math)
* [/r/learnmath](http://np.reddit.com/r/learnmath)
* [/r/mathbooks](http://np.reddit.com/r/mathbooks)
* [/r/askmath](https://np.reddit.com/r/askmath)
**Subtopics:**
#singlevariable_calculus#
Calculus is a set of mathematical techniques based on applying the idea of limit to functions, which makes it possible to study the rate at which a function changes at one specific instant rather than just its average rate of change over a finite period of time. The techniques of calculus are the foundation of physical science, and so it is no coincidence that calculus and modern physics were born simultaneously through the work of Sir Isaac Newton and his contemporaries.
**Prerequisites:**
Readers who wish to learn elementary calculus must have an understanding of arithmetic and basic algebra (manipulating algebraic expressions and solving algebraic equations). It is helpful but not necessary to be familiar with trigonometry (sine, cosine, and tangent as ratios within the unit circle and their application to geometry) and analytic geometry (parabolas, hyperbolas, conic sections, and other related functions) - these can be learned while studying the calculus.
It is important to note that *learning this topic is not nearly as difficult as its "scary" reputation might suggest*. Do not be put off by the word "calculus" - all readers who have a good grasp of basic math and basic algebra will be able to learn its techniques. Understanding the ideas behind the techniques will require you to solve many problems, think about the concepts, and eventually study theorems, but anyone can learn calculus itself. Readers should think of elementary calculus as being merely the basic grammar of science.
**Where to Start:**
Readers who wish to study calculus should pick a good introductory textbook and work through it chapter-by-chapter. These books tend to be very expensive, so readers may wish to choose a cheaper, older edition for self-study. It is very important to solve as many problems given in each section as possible - this is not just to test your reading; working (and sometimes struggling) with these problems is a necessary part of gaining proficiency in the techniques of calculus. Success will come with practice, and practice means solving problems.
At the end of a study of elementary calculus, readers should understand functions, limits, continuity, derivatives, and integrals, and should also be familiar with trigonometric, exponential, and logarithmic functions as well as sequences and series. This will prepare the reader to go on to study the mathematical laws of the physical sciences. Readers who wish to learn mathematics in more depth may wish to study analysis next, which covers the theorems and proofs behind calculus in far more depth. However, this will require an understanding of basic logic and the techniques needed to constructing proofs.
Readers who wish to study the physical sciences or engineering will discover that elementary calculus is only the first set of techniques they must master - the next steps are to learn multivariable calculus and differential equations. Multivariable calculus extends the techniques of calculus to functions of many variables (for example, one can find the volume of a geometric shape by integrating over the interior of the three-dimensional figure). This should culminate in a study of calculus applied to vector spaces, also known as vector calculus. In the study of differential equations, readers will learn how to find functions that solve equations containing derivatives - and most of the universe's rules are written in the form of differential equations.
**Books:**
* [Kleppner, Daniel and Ramsey, Norman. **Quick Calculus: A Self-Teaching Guide**. John Wiley & Sons: 1985, 2nd ed.](http://www.amazon.com/Quick-Calculus-Self-Teaching-Guide-Edition/dp/0471827223) *(a fun first tour through calculus - a good way to get a basic familiarity with the concepts, but should be followed with a more rigorous text like Larson)*
* [Kline, Morris. **Calculus: An Intuitive and Physical Approach**. Dover Publications: 1998, 2nd ed.](http://www.amazon.com/Calculus-Intuitive-Physical-Approach-Mathematics/dp/0486404536) *(focuses on intuition and the connection between calculus and science - this might be a good secondary text to help you understand why calculus is so useful)*
* [Larson, Ron. **Calculus**. Brooks Cole: 2013, 10th ed.](http://www.amazon.com/Calculus-Ron-Larson/dp/1285057090) *(An alternative to Stewart that seems to be popular with students and has many problems to solve - this might be the best place to start)*
* [Stewart, James. **Calculus**. Cengage Learning: 2012, 7th ed.](http://www.amazon.com/Calculus-7th-James-Stewart/dp/0538497815) *(the nearly-ubiquitous calculus text used in university courses, this might not be as useful as other books for self-study)*
* [Thompson, Silvanus Phillips. **Calculus Made Easy**. CreateSpace Independent Publishing Platform: 2011, 2nd reprint ed.](http://www.amazon.com/Calculus-Made-Silvanus-Phillips-Thompson/dp/1456531980) *(a conceptual explanation of calculus that can help you prepare for a textbook or provide a useful supplement; [older edition available for free on Project Gutenberg](http://www.gutenberg.org/ebooks/33283))*
**Articles:**
**Videos:**
* [Delaware's "Calculus I" lectures (UMKC)](https://www.youtube.com/playlist?list=PLF5E22224459D23D9) *(very good series of lectures that take time to explain the important techniques and concepts in depth, highly recommended for beginners)*
* [Jerison's "Single Variable Calculus" lectures (MIT)](http://ocw.mit.edu/courses/mathematics/18-01-single-variable-calculus-fall-2006/video-lectures/) *(great lectures on the important topics; you might want to watch these after working through related sections of your textbook)*
* [Khan Academy's videos on calculus](https://www.youtube.com/playlist?list=PL19E79A0638C8D449) *(a good place to look for a discussion of individual topics)*
* [Leonard's "Calculus 1" lectures](https://www.youtube.com/playlist?list=PLF797E961509B4EB5) *(explains the "nuts and bolts" of various techniques, a good place to look for worked examples)*
* [Leonard's "Calculus 2" lectures](https://www.youtube.com/playlist?list=PLDesaqWTN6EQ2J4vgsN1HyBeRADEh4Cw-)
* [patrickJMT's YouTube channel](https://www.youtube.com/user/patrickJMT/playlists) *(a good place to look for worked examples covering specific topics)*
* [Strang's "Big Picture of Calculus" (MIT)](https://www.youtube.com/watch?v=UcWsDwg1XwM) *(a short lecture explaining the concept of differentiating and integrating functions)*
* [3Blue1brown Essence of Calculus](https://www.youtube.com/playlist?list=PLZHQObOWTQDMsr9K-rj53DwVRMYO3t5Yr)
**Other Online Sources:**
* [Paul's Online Notes (Lamar)](http://tutorial.math.lamar.edu/Classes/CalcI/CalcI.aspx) *(notes are broken up by topic, includes worked exampes - a useful resource)*
* [Calculus 1 - Coursera](https://www.coursera.org/learn/calculus1) *(a free, open course in elementary calculus)*
* [List of interactive math websites at /r/learnmath](http://www.reddit.com/r/learnmath/comments/w3q6g/list_of_interactive_math_websites/)
* [/r/math](http://np.reddit.com/r/math)
* [/r/learnmath](http://np.reddit.com/r/learnmath)
* [/r/mathbooks](http://np.reddit.com/r/mathbooks)
* [/r/askmath](https://np.reddit.com/r/askmath)
**Subtopics:**
- [Multivariable Calculus](https://www.reddit.com/r/bibliographies/comments/ak9let/multivariable_calculus/)
- [Linear Algebra](https://old.reddit.com/r/bibliographies/comments/akgoky/linear_algebra/)
****
##Calculus I (Differential Calculus) [Standard Pathway Bibliography](https://old.reddit.com/r/bibliographies/comments/dc41gm/introducing_spbs/)
##Book
**Intermediate:**
* [UBC by Joel Feldman/Andrew Rechnitzer/Elyse Yeager](http://www.math.ubc.ca/%7ECLP/CLP1/clp_1_dc.pdf)
**Expert:**
* [MIT by Gilbert Strang](http://ocw.mit.edu/ans7870/resources/Strang/Edited/Calculus/Calculus.pdf)
****
##Lecture
* [Professor Leonard](https://www.youtube.com/playlist?list=PLF797E961509B4EB5)
****
##Problems
**Intermediate**
* [UBC](http://www.math.ubc.ca/%7ECLP/CLP1/problem_book_clp_1.pdf)
**Expert**
* [MIT](https://ocw.mit.edu/courses/mathematics/18-01-single-variable-calculus-fall-2005/assignments/)
****
#Multivariable calculus#
"Multivariable calculus (also known as multivariate calculus) is the extension of calculus in one variable to calculus with functions of several variables: the differentiation and integration of functions involving multiple variables, rather than just one." -Wikipedia
**Prerequisites:**
* [Basic Algebra] (https://old.reddit.com/r/bibliographies/comments/ajm8wi/basic_algebra/)
* [Precalculus] (https://old.reddit.com/r/bibliographies/comments/ajm8wi/basic_algebra/)
* [Single Variable Calculus] (https://old.reddit.com/r/bibliographies/comments/ajm8wi/basic_algebra/)
**Where to Start:**
Readers who wish to study Multivariable calculus should pick a good introductory textbook and work through it chapter-by-chapter. These books tend to be very expensive, so readers may wish to choose a cheaper, older edition for self-study. It is very important to solve as many problems given in each section as possible - this is not just to test your reading; working (and sometimes struggling) with these problems is a necessary part of gaining proficiency in the techniques of calculus. Success will come with practice, and practice means solving problems.
At the end of a study of Multivariable calculus, readers should understand Limits and Continuity, Partial Differentiation, Multiple Integration, and the Fundamental Theorem of Calculus in Three Dimensions. This will prepare the reader to go on to study the mathematical laws of the physical sciences. Readers who wish to learn mathematics in more depth may wish to study analysis next, which covers the theorems and proofs behind calculus in far more depth. However, this will require an understanding of basic logic and the techniques needed to constructing proofs.
**Books:**
* [Vector Calculus, Linear Algebra and Differential Forms: A Unified Approach](https://www.amazon.com/Vector-Calculus-Linear-Algebra-Differential/dp/0136574467)
* [Div, Grad, Curl, and All That: An Informal Text on Vector Calculus](https://www.amazon.com/Div-Grad-Curl-All-That/dp/0393925161) *A great secondary and primary book, though I would use this as a secondary book, teaches you in depth about what you need to take out of multivariable calculus.*
* [Multivariable Mathematics: Linear Algebra, Multivariable Calculus, and Manifolds](https://www.amazon.com/Multivariable-Mathematics-Algebra-Calculus-Manifolds/dp/047152638X) *Our recommended book. Good problems, a healthy explanation on what you need to know and adds in manifolds)*
* [Advanced Calculus of Several Variables (Dover Books on Mathematics)](https://www.amazon.com/Advanced-Calculus-Several-Variables-Mathematics/dp/0486683362)
* [Calculus of Several Variables (Undergraduate Texts in Mathematics)](https://www.amazon.com/Calculus-Several-Variables-Undergraduate-Mathematics/dp/0387964053)
**Videos:**
* [Professor Leonard](https://www.youtube.com/watch?v=tGVnBAHLApA&list=PLDesaqWTN6ESk16YRmzuJ8f6-rnuy0Ry7)
* [MIT OCW](https://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/)
**Problems & Exams**
* [CLP - UBC](http://www.math.ubc.ca/~CLP/CLP3/)
* [MIT](https://ocw.mit.edu/courses/mathematics/18-024-multivariable-calculus-with-theory-spring-2011/assignments/)
* [Swatchmore](http://www.swarthmore.edu/NatSci/smaurer1/Math18H/multivarH.html)
* [Duke](https://services.math.duke.edu/~bfry/math212f13/diagnostic2013.pdf)
* [Lamar](http://tutorial.math.lamar.edu/Problems/CalcIII/CalcIII.aspx)
* [Brunel](https://www.brunel.ac.uk/~mastmmb/ma2712.html)
* [Dartmouth](https://math.dartmouth.edu/~m13w17/homework.php)
* [Berkeley](https://math.berkeley.edu/~cannizzo/Fa14_Math53/53worksheets.pdf)
* [Princeton](https://www.math.princeton.edu/sites/default/files/2018-03/MAT201Sample%20Problems_1.pdf)
* [Harvard](https://scholar.harvard.edu/siams/am21a-f18-multivariable-calculus)
* [Bates](https://www.bates.edu/mathematics/math-206-multivariable-calculus-old-exams/)
* [UMD](http://schol.math.umd.edu/MVCwMATLAB/contents.html)
* [MIT](http://mit.sustech.edu/OcwWeb/Mathematics/18-02Spring-2006/Assignments/index.htm)
* [ICTM](https://www.ictm.org/resources-calculus)
* [Exster](https://www.exeter.edu/mathproblems)
**Other Online Sources:**
* [Paul's Online Notes](http://tutorial.math.lamar.edu/Classes/CalcIII/CalcIII.aspx)
* [Khan Academy](https://www.khanacademy.org/math/multivariable-calculus)
* [/r/learnmath](http://np.reddit.com/r/learnmath)
**Subtopics:**
- [Tensor Calculus](https://old.reddit.com/r/bibliographies/comments/e3csw3/tensor_calculus/)
- [Variational Calculus](https://old.reddit.com/r/bibliographies/comments/akgu7e/variational_calculus/)
* Vector Calculus
#Linear Algebra#
"Linear algebra is the branch of mathematics concerning linear equations such as linear functions such as and their representations through matrices and vector spaces. Linear algebra is central to almost all areas of mathematics. For instance, linear algebra is fundamental in modern presentations of geometry, including for defining basic objects such as lines, planes and rotations. Also, functional analysis may be basically viewed as the application of linear algebra to spaces of functions. Linear algebra is also used in most sciences and engineering areas, because it allows modeling many natural phenomena, and efficiently computing with such models. For nonlinear systems, which cannot be modeled with linear algebra, linear algebra is often used as a first-order approximation." -Wikipedia
**Prerequisites:**
* [Basic Algebra](https://old.reddit.com/r/bibliographies/comments/ajm8wi/basic_algebra/)
**Books:**
* [Linear Algebra Problem Book (Dolciani Mathematical Expositions)](https://www.amazon.com/Algebra-Problem-Dolciani-Mathematical-Expositions/dp/0883853221)
*Problem book for Linear Algebra. Highly recommended as a supplement.*
* [Introduction to Linear Algebra, Fifth Edition by Gilbert Strang](https://www.amazon.com/Introduction-Linear-Algebra-Gilbert-Strang/dp/0980232775/ref=dp_ob_title_bk) *Professor of MIT's OCW Linear Algebra Lecture series*
* [Linear Algebra and Its Applications](https://www.amazon.com/Linear-Algebra-Its-Applications-Peter/dp/0471751561)
* [Linear Algebra Done Wrong](https://www.math.brown.edu/~treil/papers/LADW/LADW.html)
*You'll get a good grip with Linear Algebra using this*
* [Linear Algebra, 4th Edition] (https://www.amazon.com/Linear-Algebra-4th-Stephen-Friedberg/dp/0130084514) *A great introductory book*
* [Introduction to Applied Linear Algebra – Vectors, Matrices, and Least Squares](https://web.stanford.edu/~boyd/vmls/)
* [Linear Algebra Done Right (Undergraduate Texts in Mathematics)] (https://www.amazon.com/dp/3319307657/ref=cm_sw_r_cp_apa_alDkAbVES094K)
*"A great book for math/physics undergrads who have already experienced matrix-centric linear algebra and would like to delve into the more abstract theory of finite-dimensional vector spaces and inner product spaces. Very clear cut with rewarding, but easy exercises." u/UglyMousanova19
* ###ADVANCED
* [Linear Algebra - As an Introduction to Abstract Mathematics](https://www.math.ucdavis.edu/~anne/linear_algebra/index.html)
* [Linear Algebra for Applications - MAS201](https://neil-strickland.staff.shef.ac.uk/courses/MAS201/)
* [Advanced Linear Algebra](https://www.springer.com/la/book/9780387728285)
**Lectures:**
* [MIT OCW](https://ocw.mit.edu/courses/mathematics/18-06-linear-algebra-spring-2010/)
* [James Hablin](https://www.youtube.com/watch?v=HAoL5fPmgrw&list=PLNr8B4XHL5kGDHOrU4IeI6QNuZHur4F86)
* [CCNY](https://www.youtube.com/watch?v=wa4xRnH-nXo&list=PLYoxM3oLTvxL_24BB4bc8kHbasgmN1eQl)
**Other Online Sources:**
* [Khan Academy](https://www.khanacademy.org/math/linear-algebra)
* [Pauls Notes](https://www.cs.cornell.edu/courses/cs485/2006sp/LinAlg_Complete.pdf)
* [3Blue1Brown](https://www.youtube.com/watch?v=fNk_zzaMoSs&list=PLZHQObOWTQDPD3MizzM2xVFitgF8hE_ab)
**Problem Sets**
* [MIT Problem sets](http://web.mit.edu/18.06/www/Spring02/probsets/)
* [MIT Problem set 2](http://web.mit.edu/18.06/www/Spring02/probsets/)
* [MIT Problem set 3](https://ocw.mit.edu/courses/mathematics/18-06-linear-algebra-spring-2010/assignments/)
* [UC Davis](https://www.math.ucdavis.edu/~linear/problem_sets.pdf)
* [PDX](https://web.pdx.edu/~erdman/LINALG/Linalg_pdf.pdf)
* [UPENN](https://www.math.upenn.edu/~kazdan/504/la.pdf)
* [Puget](http://linear.ups.edu/version3/pdf/fcla-draft-solutions.pdf) *Problems and Solutions*
**Exams**
* [Boston College Exams](https://www2.bc.edu/ilker-yuce/Spring11/MT2100102/mt210test.html)
* [MIT Exams](https://ocw.mit.edu/courses/mathematics/18-06-linear-algebra-spring-2010/exams/)
* [Bates](http://abacus.bates.edu/~etowne/205exams.htm)
* [Brandeis](http://people.brandeis.edu/~levinea/spring12/la/practicefinalsolutions.pdf)
* [Brandeis](http://people.brandeis.edu/~levinea/spring12/la/midterm1solutions.pdf)
* [Berkeley](https://math.berkeley.edu/~nadler/2014fall54midterm1sol.pdf)
* [Pitt](http://www.math.pitt.edu/~sparling/118033/118033e1.pdf)
**Captain's Log**
- Added more online sources (11/28/19)
- Added Exams, Problems and solutions (11/28/19)
#Differential Equations#
"A differential equation is a mathematical equation that relates some function with its derivatives. In applications, the functions usually represent physical quantities, the derivatives represent their rates of change, and the equation defines a relationship between the two." -Wikipedia
**Prerequisites:**
* [Single Variable Calculus](https://old.reddit.com/r/bibliographies/comments/ajm86t/singlevariable_calculus/)
* [Multivariable Calculus](https://old.reddit.com/r/bibliographies/comments/ak9let/multivariable_calculus/)
* [Linear Algebra](https://old.reddit.com/r/bibliographies/comments/akgoky/linear_algebra/)
**Books:**
* [Ordinary Differential Equations (Dover Books on Mathematics) ](https://www.amazon.com/Ordinary-Differential-Equations-Dover-Mathematics/dp/0486649407) *(Yup one book. Most Diff Eq textbooks are badly written, utilizing methods of solving Differential Equations and showing what the solutions are rather than the fundamental theorems and basis on where and why Differential Equations come from and why they do what they do. This is the only recommendation we can solely give out.)*
**Articles:**
* [Paul's Online Notes](http://tutorial.math.lamar.edu/Classes/DE/DE.aspx)
**Videos:**
* [Professor Leonard](https://www.youtube.com/watch?v=xf-3ATzFyKA&list=PLDesaqWTN6ESPaHy2QUKVaXNZuQNxkYQ_)
**Other Online Sources:**
* [MIT OCW](https://ocw.mit.edu/courses/mathematics/18-03-differential-equations-spring-2010/)
**Problems & Exams**
* [MIT](https://ocw.mit.edu/courses/mathematics/18-03-differential-equations-spring-2010/assignments/)
* [Stanford](https://math.stanford.edu/~vakil/034/index.html)
* [UCLA Problems](https://www.math.ucla.edu/~yanovsky/handbooks/PDEs.pdf)
* [Oxford](https://users.physics.ox.ac.uk/~lvovsky/yr1maths/MT/MT%207-vac%20ODE%20PS1-3%20new.pdf)
* [ResearchGate](https://www.researchgate.net/publication/332863667_PROBLEM_SET_SOLUTIONS_DIFFERENTIAL_EQUATION)
* [Swathmore College](http://www.swarthmore.edu/NatSci/wstromq1/diffeq/index.html)
* [MIT Exams](https://ocw.mit.edu/courses/mathematics/18-03-differential-equations-spring-2010/exams/)
* [PITT](http://www.math.pitt.edu/~evt3/0290/)
* [CMU](http://www.math.cmu.edu/~gheorghi/prac_sol.pdf)
* [UCDavis](https://www.math.ucdavis.edu/files/4713/7529/4837/22B-SQ08.pdf)
* [Waterloo](http://www.mhtl.uwaterloo.ca/courses/me203/exams/exam.html)
* [Purdue](https://www.math.purdue.edu/~stindel/teaching/ma266/ma266.html)
* [UNL](https://www.math.unl.edu/~mbrittenham2/classwk/221f09/exam.html)
* [BU](http://math.bu.edu/people/bob/MA226/sample-exams.html)
* [UCCS](https://www.uccs.edu/Documents/rcascava/Math3400SampleFinalExamSol.pdf)
**Subtopics:**
* [Partial Differential Equations](https://old.reddit.com/r/bibliographies/comments/akgtbv/partial_differential_equations/)
#Set Theory#
Set theory is a branch of mathematical logic that studies sets, which informally are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics. The language of set theory can be used to define nearly all mathematical objects.
**Prerequisites:**
Since set theory is a fundamental topic it doesn’t require any specific prerequisites. But, some mathematical maturity in the reader is needed to appreciate the relevance of some of the definitions and theorems. Depending on the set theory course, you may need first a course where you learn to write proofs. Or, that may be the emphasis of the set theory course itself.
**Where to Start:**
Readers who want to learn set theory should start with an introductory text book. A list of excellent choices is presented below. But, readers should be aware that there are a lot of books that teach set theory in “naive approach”. For a more complete understanding of the subject, an “axiomatic approach” must complement the naive approach. Since set theory is a first year undergraduate course, some readers might find set theory different to topics they have faced in high school. The sudden focus on proofs on seemingly “trivial” topics might be tedious for readers, but it is necessary to understand that rigorous proofs of fundamental results ensures integrity of mathematics as a whole. Readers might find taking written notes more helpful in understanding the subject.
**Books:**
* [Book of Proof](http://www.people.vcu.edu/~rhammack/BookOfProof/)
* [Elements of Set Theory](https://www.amazon.com/Elements-Set-Theory-Herbert-Enderton/dp/0122384407)
* [Naive Set Theory](https://www.amazon.com/Naive-Set-Theory-Paul-Halmos/dp/1614271313) *Stands the test of time. We recommend this to learn from.
* [The Higher Infinite: Large Cardinals in Set Theory from Their Beginnings (Springer Monographs in Mathematics)](https://www.amazon.com/Higher-Infinite-Beginnings-Monographs-Mathematics/dp/3540888667)
* [Set Theory (Studies in Logic: Mathematical Logic and Foundations)](https://www.amazon.com/Set-Theory-Studies-Logic-Mathematical/dp/1848900503) *THE graduate book to learn from*
* [Classic Set Theory: For Guided Independent Study (Chapman & Hall Mathematics S)](https://www.amazon.com/Classic-Set-Theory-Independent-Mathematics/dp/0412606100) *Another graduate book*
**Videos:**
* [Random Lectures](https://www.youtube.com/watch?v=qgizzlTVxxs&list=PLR3z46pdylqSRkO1luzzwxR6mPyTyc6ne)
**Other Online Sources:**
* [Kanamori](http://math.bu.edu/people/aki/16.pdf)
*This excellent paper by Kanamori talks about the historical development of set theory, what problems in mathematics gave rise to some of the ideas in set theory, and where the specific constructions came from. It's worth reading, or at least skimming*
* [Frederique](http://www1.spms.ntu.edu.sg/~frederique/dm4.pdf) *A short lecture notes on set theory by Frederique.*
#Real analysis#
"In mathematics, real analysis is the branch of mathematical analysis that studies the behavior of real numbers, sequences and series of real numbers, and real-valued functions. Some particular properties of real-valued sequences and functions that real analysis studies include convergence, limits, continuity, smoothness, differentiability and integrability." - Wikipedia.
**Prerequisites:**
* [Single Variable Calculus](https://old.reddit.com/r/bibliographies/comments/ajm86t/singlevariable_calculus/)
* [Multivariable Calculus](https://old.reddit.com/r/bibliographies/comments/ak9let/multivariable_calculus/)
* [Proof Techniques](https://old.reddit.com/r/bibliographies/comments/ajq34w/proof_techniques/)
**Books:**
* [A First Course in Real Analysis](http://www.amazon.com/First-Course-Analysis-Undergraduate-Mathematics/dp/0387974377/ref=sr_1_1?ie=UTF8&s=books&qid=1273347024&sr=1-1) *Our recommendation for analysis*
* [Walter Rudin's *Principles of Mathematical Analysis*](https://notendur.hi.is/vae11/%C3%9Eekking/principles_of_mathematical_analysis_walter_rudin.pdf) *The god level book of analysis*
* [*Real Analysis and Applications*](http://www.springer.com/us/book/9780387980973)
* [Real Mathematical Analysis by Pugh](https://www.amazon.com/Mathematical-Analysis-Undergraduate-Texts-Mathematics/dp/144192941X).
* [Tao, *Analysis I*](https://terrytao.wordpress.com/books/analysis-i/)
* [Elementary Analysis Calculus ](http://www.amazon.com/Elementary-Analysis-Calculus-Kenneth-Ross/dp/038790459X)
* [Buck, Advanced Calculus](http://www.math.harvard.edu/~shlomo/docs/Advanced_Calculus.pdf)
**How to Learn:**
* [MIT](https://ocw.mit.edu/courses/mathematics/18-100c-real-analysis-fall-2012/) *Full course*
* [Trinity](http://ramanujan.math.trinity.edu/wtrench/texts/TRENCH_REAL_ANALYSIS.PDF) *Lecture Notes*
* [University Of Louisville](http://www.math.louisville.edu/~lee/RealAnalysis/) *More notes*
* [University of Hawaii](http://www.math.hawaii.edu/~williamdemeo/Analysis-nohref.pdf) *Problems and Solutions*
* [Temple University](https://math.temple.edu/~gutierre/math8041/problems) *Problems*
* [University of Georgia](https://www.math.uga.edu/real-analysis-exams) *Exams*
* [University of New Mexico](https://math.unm.edu/graduate/past-qualifying-exams-real-analysisreal-variables) *Exams*
* [MSU](https://users.math.msu.edu/users/hendricks/Math131A/Math%20131A%20Sample%20Final%201.pdf) *Exams*
* [WUSTL](https://www.math.wustl.edu/~wick/teaching/math4317/math4317_exam1.pdf) *Exams*
* [UCDavis](https://www.math.ucdavis.edu/~hunter/m125a/m125a_sample_final_solutions.pdf) *Exams*
* [UCSD](http://math.ucsd.edu/~doprea/140f14/finalp.pdf) *Exams*
**Lectures:**
* [Harvey Mudd](https://www.youtube.com/watch?v=sqEyWLGvvdw&list=PL0E754696F72137EC)
* [Bethel University](https://www.youtube.com/watch?v=EaKLXK4hFFQ&list=PLmU0FIlJY-MngWPhBDUPelVV3GhDw_mJu)
* [IIT Madras](https://www.youtube.com/watch?v=md5UCR7mcIY&list=PLbMVogVj5nJSxFihV-ec4A3z_FOGPRCo-)
* [Scripps College](https://www.youtube.com/watch?v=8hJYD_NcOGQ&list=PLun8-Z_lTkC5HAjzXCLEx0gQkJZD4uCtJ)
**Subtopics:**
- [Complex Analysis](https://old.reddit.com/r/bibliographies/comments/axuhxy/complex_analysis/)
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This was posted by a user, whom I've banned due to being active participant in a quarantined community.
[George Bergman's companion exercises to Rudin's textbook for Chapters 1-7](https://math.berkeley.edu/~gbergman/ug.hndts/m104_Rudin_exs.pdf).
[Roger Cooke's solutions manual for Rudin's analysis](https://minds.wisconsin.edu/handle/1793/67009)
[A subreddit devoted to Baby Rudin with further resources in the sidebar.](https://www.reddit.com/r/babyrudin/)
[Tom Apostol's textbook](https://www.amazon.com/Mathematical-Analysis-Second-Tom-Apostol/dp/0201002884)
I find that Rudin is to Analysis textbooks what C++ is to programming languages. A little difficult at first, but with so many auxiliary sources that it becomes one of the best texts to learn from in spite of this.
#Partial Differential Equations#
In mathematics, a partial differential equation is a differential equation that contains beforehand unknown multivariable functions and their partial derivatives. PDEs are used to formulate problems involving functions of several variables, and are either solved by hand, or used to create a computer model. -Wikipedia
**Prerequisites:**
* [Multivariable Calculus](https://old.reddit.com/r/bibliographies/comments/ak9let/multivariable_calculus/)
* [ODE](https://old.reddit.com/r/bibliographies/comments/akgt8r/differential_equations/)
* [Real Analysis](https://old.reddit.com/r/bibliographies/comments/axuhu3/real_analysis/)
**Books:**
* [Partial Differential Equations for Scientists and Engineers (Dover Books on Mathematics)](https://www.amazon.com/Differential-Equations-Scientists-Engineers-Mathematics/dp/048667620X/ref=pd_lpo_sbs_14_t_1?_encoding=UTF8&psc=1&refRID=HHMZ1C0CG2M2X6D0WR4R )*(A great starter book)*
* [Partial Differential Equations: Second Edition (Graduate Studies in Mathematics) 2nd Edition
](https://www.amazon.com/Partial-Differential-Equations-Graduate-Mathematics/dp/0821849743)
* [Partial Differential Equations: An Introduction 2nd Edition](https://www.amazon.com/Partial-Differential-Equations-Walter-Strauss/dp/0470054565)
* [Partial Differential Equations (Applied Mathematical Sciences) (v. 1) 4th Edition
](https://www.amazon.com/Partial-Differential-Equations-Mathematical-Sciences/dp/0387906096)
* [Partial Differential Equations: Methods and Applications](https://www.amazon.com/Partial-Differential-Equations-Methods-Applications/dp/0130093351) *(by McOwen is a good book, shorter and easier than Evans (and also making use of distributions)).*
**Videos:**
* [Lamar] (http://www.math.lamar.edu/faculty/maesumi/PDE1.html#Lectures)
* [ICTP](https://www.youtube.com/watch?v=Rq1iRT2LL-8&list=PLLq_gUfXAnkkvL_UoCGivS0wOYhwCtczI)
* [MIT OCW](https://ocw.mit.edu/courses/mathematics/18-152-introduction-to-partial-differential-
equations-fall-2011/)
**Other Online Sources:**
* [UCLA](https://www.math.ucla.edu/~yanovsky/handbooks/PDEs.pdf)
* [Leipzig](https://www.math.uni-leipzig.de/~miersemann/pdebook.pdf)
* [Unkown](http://issc.uj.ac.za/downloads/problems/partial.pdf)
* [Princeton](https://web.math.princeton.edu/~seri/homepage/papers/gws-2006-3.pdf)
* [UCSB](http://web.math.ucsb.edu/~grigoryan/124A.pdf)
* [PSU](http://www.personal.psu.edu/sxt104/class/Math251/Notes-PDE%20pt1.pdf)
* [UTAH](https://www.math.utah.edu/~alali/SolutionsManual.pdf)
* [Homework](http://www.math.lamar.edu/faculty/maesumi/PDE/hwpde09.pdf)
* [Resources](http://www.math.lamar.edu/faculty/maesumi/PDE1.html#pdeRESOURCES)
#Calculus of Variations#
Calculus of variations is a field of mathematical analysis that uses variations, which are small changes in functions and functionals, to find maxima and minima of functionals: mappings from a set of functions to the real numbers.- Wikipedia
**Prerequisites:**
* [Multi-variable Calculus](https://old.reddit.com/r/bibliographies/comments/ak9let/multivariable_calculus/)
* [ODE](https://old.reddit.com/r/bibliographies/comments/akgt8r/differential_equations/)
* [PDE](https://old.reddit.com/r/bibliographies/comments/akgtbv/partial_differential_equations/)
**Books:**
* [Calculus of Variations (Dover Books on Mathematics)](https://www.amazon.com/Calculus-Variations-Dover-Books-Mathematics/dp/0486414485) *We reccomend using this textbook until hitting Landau in Physics.*
* [Applied Mathematical Methods in Theoretical Physics 2nd Edition
](https://www.amazon.com/Applied-Mathematical-Methods-Theoretical-Physics/dp/352740936X/ref=la_B001JOU43Y_1_1?s=books&ie=UTF8&qid=1390327904&sr=1-1)
* [Calculus of Variations (Dover Books on Mathematics)](https://www.amazon.com/Calculus-Variations-Dover-Books-Mathematics-ebook/dp/B00EYLI70G/ref=sr_1_5?keywords=variational+calculus&qid=1551653801&s=books&sr=1-5)
* [Calculus of Variations: An Introduction to the One-Dimensional Theory with Examples and Exercises (Texts in Applied Mathematics)](https://www.amazon.com/Calculus-Variations-Introduction-One-Dimensional-Mathematics/dp/3319711229/ref=sr_1_7?keywords=variational+calculus&qid=1551653801&s=books&sr=1-7)
**Articles:**
* [Lecture Notes](http://www.staff.city.ac.uk/~george1/calculus-variations-optimal-control.pdf)
* [Lecture Notes](https://math.cmu.edu/CNA/Publications/publications2016/papers/16-CNA-007.pdf)
**Videos:**
* *(Due to the nature of this field being a subset of Optimization Theory, there aren't any full online courses at this time)*
#Topology#
#**Brief Explanation**
In mathematics, topology is concerned with the properties of space that are preserved under continuous deformations, such as stretching, twisting, crumpling and bending, but not tearing or gluing. An n-dimensional topological space is a space with certain properties of connectedness and compactness. - Wikipedia
**Prerequisites:**
* [Multivariable Calculus](https://old.reddit.com/r/bibliographies/comments/ak9let/multivariable_calculus/)
* [Set theory](https://old.reddit.com/r/bibliographies/comments/aljhaw/set_theory/)
**Books:**
* [Topology (2nd Edition) ](https://www.amazon.com/Topology-2nd-James-Munkres/dp/0131816292) *The best written book for topology hands down*
* [General Topology (Dover Books on Mathematics) ](https://www.amazon.com/General-Topology-Dover-Books-Mathematics/dp/0486434796 ) *Covers some information not in Munkres*
* [ A First Course in Topology: Continuity and Dimension] (https://pages.vassar.edu/mccleary/books/a-first-course-in-topology-continuity-and-dimension/) *A great introduction book*
* [Elements of Combinatorial and Differential Topology (Graduate Studies in Mathematics, Vol. 74)](https://www.amazon.com/Elements-Combinatorial-Differential-Topology-Mathematics/dp/0821838091) *A different approach, using more geometric methods*
* [Milnor, Topology from a Differentiable Viewpoint] (https://www.maths.ed.ac.uk/~v1ranick/papers/milnortop.pdf) *Assumes no pre-requiste knowledge*
**Articles**
* [Curlie](https://curlie.org/Science/Math/Topology)
* [Notes](https://www.math.colostate.edu/~renzo/teaching/Topology10/Notes.pdf)
* [Glossary](https://web.archive.org/web/20090801231406/http://www.ornl.gov/sci/ortep/topology/defs.txt)
**Problems & Exams**
* [PDRMI](http://www.pdmi.ras.ru/~olegviro/topoman/eng-book-nopfs.pdf)
* [STEMZ](https://stemez.com/subjects/maths/1PTopology/1PTopology.php)
* [iitk](http://home.iitk.ac.in/~chavan/topology_mth304.pdf)
* [PSU](http://www.personal.psu.edu/axk29/Pr-exams94.pdf)
* [Arizona](https://www.math.arizona.edu/files/grad/workshops/integration/topology_probs_2019.pdf)
* [Topology Exams](https://www.math.uga.edu/topology-exams)
**Videos:**
* [MIT OCW](https://ocw.mit.edu/courses/mathematics/18-901-introduction-to-topology-fall-2004/)
* [African Institute for Mathematical Sciences (South Africa)](https://www.youtube.com/watch?v=SXHHvoaSctc&list=PLTBqohhFNBE_09L0i-lf3fYXF5woAbrzJ)
* [ICTP](https://www.youtube.com/watch?v=PXIcas22MtQ&list=PLLq_gUfXAnkl8bjQh-hGQ9u24xZP9x0dx)
**Subtopics**
* Algebraic Topology
**Captain's Log**
* Added more problems (11/29/2019)
#Differential Geometry#
"Differential geometry is a mathematical discipline that uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra to study problems in geometry." -Wikipedia
**Prerequisites:**
* [Multivariable Calculus](https://old.reddit.com/r/bibliographies/comments/ak9let/multivariable_calculus/)
* [Linear Algebra] (https://old.reddit.com/r/bibliographies/comments/akgoky/linear_algebra/)
* [Topology] (https://old.reddit.com/r/bibliographies/comments/akguwi/topology/)
**Books:**
* [Vector Analysis (Undergraduate Texts in Mathematics)](https://www.amazon.com/Vector-Analysis-Undergraduate-Texts-Mathematics/dp/0387986499) *Our recommendation for an intro book*
* [Geometric Integration Theory (Cornerstones)] (https://www.amazon.com/Geometric-Integration-Theory-Cornerstones-Steven/dp/0817646760)
* [Introduction to Smooth Manifolds (Graduate Texts in Mathematics)](https://www.amazon.com/Introduction-Smooth-Manifolds-Graduate-Mathematics/dp/0387954481)
* [Differential Geometry (Wiley Classics Library)](https://www.amazon.com/Differential-Geometry-Wiley-Classics-Library/dp/0471504033) *A great intro book*
* [Elementary Differential Geometry](https://www.amazon.com/Elementary-Differential-Geometry-Christian-B%C3%A4r/dp/0521721490) *Not the greatest book, but does things right*
**Exams**
* [ Math 405/538 Differential Geometry
Final Exam](https://www2.bc.edu/baris-coskunuzer/courses/math405_F13/Math405final_F13_key.pdf)
* [Math 352] (http://faculty.bard.edu/belk/math352/)
* [Final Exam of Differential Geometry I ] (http://web.ntnu.edu.tw/~494402345/differential_geometry/2008final_differential_geometry.pdf)
* [Final Exam, Math 421