As the title says, I would like to self-study multivariable real analysis (integration) 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 Fubinis 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
Diferential forms
Stokes' theorem and it's 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".
Please let me know if you need more information or clarifications. Thanks in advance for your replies.
EDIT:
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.
EDIT 2:
I also speak German so suggestions of videos and books in German are also welcome.
I think the book Advanced calculus by Loomis, Sternberg would suit your needs. Its freely available from the website of the second author.