# Reference request for self-studying multiple Riemann integrals

• MHB
• AltairAC
In summary, a Riemann integral is a type of integral used to calculate the area under a curve, developed by mathematician Bernhard Riemann. To self-study multiple Riemann integrals, a strong understanding of single variable integrals and multivariable calculus is necessary, along with resources such as textbooks and online lectures. The difference between single and multiple Riemann integrals is that the latter is used to calculate volume in multiple dimensions and requires more complex calculations. Recommended resources for self-studying include textbooks and online courses, and knowledge of multiple Riemann integrals can be applied in various real-world situations such as physics, engineering, and economics.
AltairAC
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 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:

(i) 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 $\ \mathbb{R}^2$.(ii) 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.

EDIT:
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? ( (i) and (ii) )

Last edited:

Thank you for your interest in self-studying multivariable real analysis. It is a challenging subject, but with dedication and the right resources, you can definitely succeed.

Based on your background and the topics covered in your notes, I would recommend the following books and resources:

1. "Calculus: Early Transcendentals" by James Stewart - This textbook covers multivariable calculus, including topics such as double and triple integrals, change of variables, and vector calculus. It also has many examples and detailed explanations, making it suitable for self-study. It is available in both English and German.

2. "Advanced Calculus: A Differential Forms Approach" by Harold M. Edwards - This book covers multivariable calculus using the language of differential forms, which can be helpful in understanding the concepts of integration and differentiation in higher dimensions. It also has a lot of examples and exercises for practice.

3. "Calculus on Manifolds" by Michael Spivak - This is a more advanced textbook that covers multivariable calculus and differential forms in a rigorous and theoretical manner. It may be challenging for self-study, but it is an excellent resource for a deeper understanding of the subject.

In addition to these books, I would recommend watching lectures on multivariable calculus and real analysis on platforms like YouTube and Coursera. Some good channels to check out are MIT OpenCourseWare and Khan Academy.

Lastly, I would also suggest supplementing your studies with practice problems and exercises from textbooks or online resources. This will help solidify your understanding of the concepts and prepare you for exams.

I hope these recommendations are helpful to you. Good luck with your self-study journey!

## 1. What is a Riemann integral?

A Riemann integral is a type of integral that is used to calculate the area under a curve. It was developed by the mathematician Bernhard Riemann and is a fundamental concept in calculus.

## 2. How do I self-study multiple Riemann integrals?

To self-study multiple Riemann integrals, it is important to have a strong understanding of single variable Riemann integrals and multivariable calculus. It is also helpful to have a textbook or online resources that provide practice problems and explanations.

## 3. What is the difference between a single variable Riemann integral and a multiple Riemann integral?

A single variable Riemann integral is used to calculate the area under a curve in one dimension, while a multiple Riemann integral is used to calculate the volume under a surface in two or more dimensions. Multiple Riemann integrals also require the use of double or triple integrals and may involve more complex calculations.

## 4. Can you recommend any resources for self-studying multiple Riemann integrals?

There are many resources available for self-studying multiple Riemann integrals, including textbooks, online lectures, and practice problems. Some recommended textbooks include "Calculus: Early Transcendentals" by James Stewart and "Multivariable Calculus" by James Stewart. Khan Academy and MIT OpenCourseWare also offer free online courses on multivariable calculus.

## 5. How can I apply my knowledge of multiple Riemann integrals in real-world situations?

Multiple Riemann integrals are commonly used in physics, engineering, and economics to calculate volumes, areas, and other quantities in real-world scenarios. For example, they can be used to calculate the mass of an irregularly shaped object, the surface area of a 3D shape, or the work done by a force on an object. Understanding multiple Riemann integrals can also help in solving optimization problems and in understanding multivariable functions.

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