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I preserve numbering as much as possible. So for example exercises are only added with new numbers, so that old exercises are not renumbered, etc... I try to preserve pagination as well though adding a page in the middle is sometimes unavoidable.

**May 7th 2018 edition, Version 2.0 (edition 2, 0th update):**

Numbering of definitions, examples, propositions changed in 8.1, 8.3, 10.1.
**Numbering of exercises is unchanged, except for 9.1.7 which was replaced
due to erratum.**

**New section 10.7**on change of variables.**New chapter 11**on Arzela-Ascoli, Stone-Weierstrass, power series, and Fourier series.- A
**List of Notations**is added at the back as in volume I. - In the PDF the pages have been made slightly longer so that we can lower the page count to save some paper.
- Add figure showing vector as an arrow and discussion about this for those that do not remember it from vector calculus.
- Add a paragraph about simple algebraic facts such that $0v=v$.
- Add footnote about linear independence for arbitrary sets in 8.1.
- Add example that span of $t^n$ is ${\mathbb{R}}[t]$.
- Add remark about proving a set is a subspace.
- We also use the words "linear operator" for $L(X,Y)$, and it is for $L(X)$ that we say "linear operator on $X$", so update the definition appropriately.
- Add convexity of $B(x,r)$ as a proposition since we use it so often.
- Add exercise 8.1.19
- Proposition 8.2.4 doesn't need $Y$ to be finite dimensional, same in the exercise 8.2.12, so no need to assume it.
- In Proposition 8.2.5, emphasize where the finiteness of dimension is needed.
- Use $GL(X)$ as notation for invertible linear operators.
- Give more detail on why mapping between matrices and linear operators is one to one once a basis is fixed.
- Add a commutative diagram to the independence on basis discussion.
- Reorder the definition of sign of a permutation to be more logical.
- Add short example of permutation as transpositions.
- Add exercises 8.2.14, 8.2.15, 8.2.16, 8.2.17, 8.2.18, 8.2.19
- Add figure to definitions 8.3.1 and 8.3.8.
- Add Proposition 8.3.6, which was conspicuously missing.
- Add figure for differentiable curve and its derivative.
- Add figure to exercises 8.3.5 and 8.3.6.
- Add exercise 8.3.14
- Add graph to figure in example 8.4.3 (and adjust the formulas)
- As application of continuous partials imply $C^1$
- Add exercises 8.4.7, 8.4.8, 8.4.9, 8.4.10
- Fix up statement of the inverse function theorem in 8.4.
- Add a couple of figures to proof of the inverse function theorem.
- Add a figure to the implicit function theorem.
- Add a short paragraph about the famous Jacobian conjecture.
- make the remark at the end of 8.5 into an actual "remark"
- Add observation about solving a bunch of equations not just for $s=0$ for the implicit function theorem.
- Add exercises 8.5.9, 8.5.10, 8.5.11
- Add figure to 8.6
- In 8.6 cleanup the argument in the proposition and use only positive $s$ and $t$ for simplicity.
- Add exercises 8.6.5, 8.6.6, 8.6.7
- Refer to the new proposition 7.5.12 about the continuity in 9.1
- Add figure to example in 9.1
**Exercise 9.1.7 replaced due to erratum**. The replacement shows the same issue that the previous wrong exercise tried to.- Add exercise 9.1.8
- Reorder the introduction of 9.2 a bit, and fix an erratum in that derivative at the endpoints was not really defined for mappings.
- Add figure to examples 9.2.2, 9.2.3, 9.2.11, 9.2.13, 9.2.18
- Add figure for definition of a function against arc-length measure.
- Add figure to proof of path independence implies antiderivative in 9.3.
- Add figure to proof that integral over closed paths being zero means that the integral is path independent in 9.3.
- Add figure to definition 9.3.5
- Change hint to exercise 9.3.8.
- Add Example 10.1.16 of compact support with a figure, following examples/propositions in 10.1 are renumbered.
- Explicitly mention monotonicity of outer measures right after the definition (it is a rather easy exercise), and also allowing finite sequences of rectangles in the definition (a new exercise).
- Add figure to definition of outer measure.
- Clean up proof of Proposition 10.3.2.
- Add exercises 10.3.11, 10.3.12 (and a figure), 10.3.13.
- Add corollary for the Riemann integrability theorem showing that it is an algebra, that min and max of two functions are Riemann integrable and so is the absolute value.
- Add exercises 10.4.6, 10.4.7, 10.4.8, 10.4.9, 10.4.10, 10.4.11
- Add exercises 10.5.5, 10.5.6, 10.5.7
- In 10.6, add figure for positive orientation and a figure illustrating the three types of domains.
- Many minor improvements in style and clarity, plus several small new example throughout.
- Fix errata.

**March 21st 2017 edition, Version 1.0 (edition 1, 0th update):**

First version.