# Self-studying multiple Riemann integrals

#### Altair

As the title says, I would like to self-study multivariable real analysis (integration, specifically; the Riemann integral) and I need some recommendations (resources, books, videos, ...).

I'm from Croatia and got my hands on some Croatian notes about multivariable real analysis so if some of the things I mention don't make sense, please let me know and I'll try to clarify. The notes I got aren't suitable for self-study, but I thought it might be useful to mention what they contain.

The notes start of with a review of the single variable case (Darboux sums, properties of the Riemann integral). Then we look at a bounded function $f:[a,b]\times[c,d]\rightarrow \mathbb{R}$ and define the appropriate Darboux sums and integral. Very often, it is emphasized that it is important that the domain is a rectangle whose sides are parallel with the coordinate axes. After that, the notes deal with Fubini's theorem. Then the notes deal with some properties of Darboux sums:

* Every lower Darboux sum is smaller than every upper Darboux sum.
* A bounded function $f:A=[a,b]\times[c,d]\rightarrow \mathbb{R}$ is integrable on A iff $\forall \epsilon > 0 \ \exists$ subdivision P of the rectangle A so that $S(P)-s(P) <\epsilon$

After that:

- Areas of sets in $\mathbb{R}^{2}$.

- Proof of Lebesgue's theorem (something about oscilations)
$$O(f,c) = \inf _{c\in U} \sup _{x_1,x_2 \in U \cap A} | f(x_1) - f(x_2)|$$

- Properties of the double integral (linearity, ...)

- Change of variables in a double integral $\int_{D} f = \int_{C} (f \circ \phi) \cdot | J_{\phi} |$

- Integral sums and Darboux's theorem

- Functions defined via integral $F(y) = \int_{a}^{b} f(x,y) dx$

- Multiple integrals (n-dimensional domain)

- Integrals of vector functions

- Smooth paths

- Integral of the first kind

- Integral of the second kind and differential 1-forms

- Green's theorem

- Multilinear functions

- Areas of surfaces

- Differential forms

- Stokes' theorem and its applications

- Classical theorems of vector analysis (Gauss' theorem - divergence theorem?, classical Stokes' theorem, ...)

Since it's for self-study, it would be cool if the books (videos, ...) contained detailed proofs and examples because I want to be able to make valid arguments for claims such as these:

The notes I've got ask such questions as "Does a disk have an area?", "Does a triangle have an area?" where area is defined as:

Definition. We say that C has an area if the function $\chi _C$ is integrable on C, i.e. on some rectangle that contains C. In that case, the area of C is $\nu (C) = \int _C \chi _C$ where:

$\chi _C (x,y) = \begin{cases} 1, (x,y) \in C \\ 0, (x,y) \notin C \end{cases}$

and C is a bounded subset of $\ \mathbb{R}^2$.

Another example:
$C =\{ (x,x) | x\in\mathbb{R} \}$

C has a (Lebesgue) measure of zero.
The notes say that the argument "C is just a rotated x-axis" is not valid because $d(k , k+1) = (k+1) - k = 1 < d(f(x_{k_1}), f(x_k))$ so we have a rotation and "stretching".

My background: I've got a good understanding of real analysis in one variable ($\epsilon - \delta$ proofs, sequences, continuity and differentiability of real functions of a real variable, the definite and indefinite Riemann integral of functions in one variable (Darboux sums), Taylor series). I'm familiar with the following concepts in $\mathbb{R}^n$: open, closed and compact sets, sequences and limits, connectedness and path connectedness, continuity and differentiability of real multivariable functions, local extrema and the mean value theorem. I also speak German, so suggestions of videos and books in German are also welcome.

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#### Altair

The book Elementary Classical Analysis by Jerrold E. Marsden and Michael J. Hoffman seems to be the kind of book I am looking for. Are there any similar books available, preferably books that contain examples like the ones I mentioned above?

#### [email protected]

The absolute best book(s) on multivariable analysis is Kolk and Duistermaat:
https://www.amazon.com/Multidimensional-Real-Analysis-Differentiation-Mathematics/dp/0521551145

It is extremely!! thorough, and it does things the best way possible.

Half of the book are devoted to problem statements. And the problems are very well chosen. No mindless computations from calculus, but rather tough (but not TOO hard) proofy questions. If you want to master multivariable, this is the book to go to.

Second, there is Hubbard, Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach
Be sure to go for the 4th or 5th edition.
Less rigorous at times, but an amazing introduction to differential forms which are explained very intuitively. Best introduction I've seen. The book also contains an intro to Electromagnetism from a forms point of view.

The standard is of course Spivak:
https://www.amazon.com/Calculus-Manifolds-Approach-Classical-Theorems/dp/0805390219
I consider it to be too dense and too short to be really useful. But I do think the exercises are particularly well chosen.