📋 Changes in this PR: Equalities, Inequalities and Minimizers, Complex Numbers, Polynomials, Exponentials and Logarithms, Convexity, Infimum and Sums, Trigonometry, Algebra, Derivatives and Taylor Series, Double Integrals and Fubini’s Theorem, Integrals and the Fundamental Theorem of Calculus, Calculus, Dependency Tree, Expressions and Algorithms, Functions, Growth Rates, Math for Data Science, Indicator Functions, Induction and Recursion, Complex Vectors and Matrices, Diagonalization and Cholesky, Direct Sums and Orthogonal Complements, Inner Products and Orthogonality, Matrices, Projections and Linear Systems, Column Space, Null Space and Rank-Nullity, Special Matrices and Decompositions, Subspaces and Rank, Vectors, Linear Algebra, Additivity, Measurable Functions and Integrals, \(L^p\) Spaces, Measures and Probability Measures, Product Measures, Sigma-Algebras, Measures, Notation, Number Systems, Quadratic Minimization under Linear Constraints, Convexity in Several Variables, Critical Points and Optimality Conditions, Gradient Descent, Newton’s Method, Quasi-Newton Methods, Optimization, Proof Writing, Relations, Sets and Functions, Sets, Vector Calculus
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Calculus

The calculus notes are split across these pages:

• Derivatives and Taylor Series: limits and derivatives, derivative rules, tangent lines, linear approximation, and Taylor series.
• Integrals and the Fundamental Theorem of Calculus: antiderivatives, regularity conditions, and the Fundamental Theorem of Calculus.
• Double Integrals and Fubini’s Theorem: double integrals over rectangles and non-rectangular regions, and Fubini’s and Tonelli’s theorems.

A link to a section, theorem or definition on this page from before the split goes on to the page that now holds it.

1 Further reading

• Kaplan [1]
• Khuri [2]
• Banner [3]
• Larson and Edwards [4]
• Miller [5] The calculus review exercises and worked solutions across these pages and in Vector Calculus are adapted from this supplemental review chapter.
  • http://www.youtube.com/watch?v=xYzQL0TUtBA
  • http://www.youtube.com/watch?v=Ps2SBo_WjoE
• Grinberg [6] (the rigorous foundations behind these results)
• Essence of Calculus is a 12-video YouTube playlist by Grant Sanderson (3Blue1Brown) that develops geometric and visual intuition for limits, derivatives, product and chain rules, implicit differentiation, integration, and Taylor series.

References

[1]

D. Kaplan, MOSAIC calculus. www.mosaic-web.org, 2022. Available: www.mosaic-web.org

[2]

A. I. Khuri, Advanced calculus with applications in statistics. John Wiley & Sons, 2003. doi: 10.1002/0471394882.

[3]

A. D. Banner, The calculus lifesaver : All the tools you need to excel at calculus. in A princeton lifesaver study guide. Princeton, New Jersey: Princeton University Press, 2007. Available: https://press.princeton.edu/books/paperback/9780691130880/the-calculus-lifesaver

[4]

R. Larson and B. H. Edwards, Calculus, 11th ed. Boston, MA: Cengage Learning, 2018. Available: https://www.cengage.com/c/calculus-11e-larson/

[5]

S. J. Miller, The probability lifesaver: Calculus review problems. 2016. Available: https://web.williams.edu/Mathematics/sjmiller/public_html/probabilitylifesaver/index.htm#:~:text=http%3A//web.williams.edu/Mathematics/sjmiller/public_html/probabilitylifesaver/supplementalchap_calcreview.pdf

[6]

R. Grinberg, The real analysis lifesaver: All the tools you need to understand proofs, 1st ed. in Princeton lifesaver study guides. Princeton: Princeton University Press, 2017. Available: https://press.princeton.edu/books/paperback/9780691172934/the-real-analysis-lifesaver