General Analysis. A guide to the frameworks of calculus, complex analysis and functional analysis consisting of standard theorems and proofs as well as some original ideas.
Differentiation in Banach spaces. An introduction to the Fréchet derivative and its most useful properties.
- Differentiation done correctly: 1. The derivative
- Differentiation done correctly: 2. Higher derivatives
- Differentiation done correctly: 3. Partial derivatives
- Differentiation done correctly: 4. Inverse and implicit functions
- Differentiation done correctly: 5. Maxima and minima
- Convex functions, second derivatives and Hessian matrices
- Fréchet derivative of the (matrix) exponential function
- First-order ODEs, matrix exponentials, and det(exp)
Line integrals in Banach spaces. A general formulation of the line integral, and an introduction to complex analysis from a more advanced point of view.
- Line integrals: 1. Exact, conservative and closed forms
- Line integrals: 2. Locally exact forms and singular homology
- Line integrals: 3. Applications to complex analysis
- Horn’s inequality for singular values via exterior algebra
- Every continuous open mapping of R into R is monotonic
- Formula for the circumference of an ellipse
- Power series of tan, cot and csc
I’ve studied some mathematics on my own; on this page are books that I have read along with some comments. Please note that I cannot guarantee the mathematical validity/correctness/accuracy of the content below.
John M. Lee’s Introduction to Smooth Manifolds. Click here for my (very incomplete) solutions.
Topics: Smooth manifolds.
Prerequisites: Algebra, basic analysis in Rn, general topology, basic algebraic topology.
John M. Lee’s Introduction to Topological Manifolds. Click here for my solutions.
Topics: General topology, algebraic topology.
Prerequisites: Metric spaces and basic group theory, but no general topology.
James Munkres’s Analysis on Manifolds.
Topics: Analysis in Rn, vector calculus, smooth submanifolds of Rn.
Prerequisites: Basic real analysis, linear algebra.
Serge Lang’s Complex Analysis.
Topic: Complex analysis.
Prerequisites: Basic real analysis.
Frederick M. Goodman’s Algebra: Abstract and Concrete. (Free!)
Topic: (Abstract) algebra.
Here are some other great books: