Learning the theory of the n-dimensional Riemann integral

In summary, The conversation discusses the speaker's desire to learn the theory behind the n-dimensional Riemann integral and their difficulties finding suitable materials for self-study. They mention finding lecture notes and asking for recommendations for books, notes, and video lectures in Croatian, English, or German. The conversation also mentions two examples, one about defining an area for a bounded subset of ##\mathbb{R}^2 ## and the other about the Lebesgue measure of a specific set. The conversation also includes recommendations for free books on the topic, and a mention of measure theory and its properties compared to Riemann integration.
  • #1
AltairAC
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1
I would like to learn (self-study) the theory behind the n-dimensional Riemann integral (multiple Riemann integrals, not Lebesgue integral). I am from Croatia and found lecture notes which Croatian students use but they are not suitable for self-study. The notes seem to be based on the book: J. E. Marsden, M. J. Hoffman, Elementary Classical Analysis but the book is expensive. Could you recommend me some books/notes/video lectures/... (languages: Croatian, English, German) ?

I also got some student notes and while going through them, two examples caught my eye. I want to learn how to make valid mathematical arguments for such examples.

Example 1.
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 ##.

The notes I've got ask such questions as "Does a disk have an area?", "Does a triangle have an area?". (in terms of the definition above)

Example 2.
$$ C =\{ (x,x) | x\in\mathbb{R} \} $$
Claim: 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".

Thanks in advance for your answers!
 
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well i am puzzled that saying C is a rotated x-axis is not valid. it certainly is a rotated x axis, but the map taking x to (x,x) does not happen to be the rotation map. still there is a rotation map from the axis onto this set.

second, there are free books available which explain the n dimensional "Riemann" integral, although just what that means may be variable since Riemann himself did not treat several variable integration, at least not in his famous paper on trigonometric series, where he defined his (one variable) integral.

The book Advanced Calculus by Loomis and Sternberg, although too abstract in many ways for convenient use, may be suitable for its theory of n dimensional "content" and integration (Riemann integration, not Lebesgue), and was free on Sternberg's website at Harvard, last time I looked.

The difference between Riemann and Lebesgue integration, or between content and measure, is the use of either finitely many or infinitely many rectangles to cover sets. For instance if you try to cover the rational points of the unit interval with a finite number of rectangles, their total length must be ≥ 1, but if you use an infinite number you can make the total length as short as you wish. So those rationals have content 1 but measure zero.

yes here it is: look at chapter 8:

http://www.math.harvard.edu/~shlomo/docs/Advanced_Calculus.pdf
 
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  • #3
I would highly recommend measure theory if you’re up for it - the Lebesgue Theory is very nice and has many better properties than the Riemann integral.

However, since you did say you wanted to learn Riemann integration on Euclidean space, I’d recommend Duistermaat and Kolk’s Multidimensional Analysis, Volume 2.
 
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  • #4
I think most Advanced Calc books contain this material.
 
  • #5
WWGD said:
I think most Advanced Calc books contain this material.

Yeah, most treat it very badly. E.g. Rudin.
 
  • #6
Math_QED said:
Yeah, most treat it very badly. E.g. Rudin.
But Rudin's is an Analysis book, not a book on Advanced Calculus.
 

1. What is the n-dimensional Riemann integral?

The n-dimensional Riemann integral is a mathematical concept used to calculate the volume or area of a function in n-dimensional space. It is an extension of the one-dimensional Riemann integral, which is used to calculate the area under a curve on a single axis.

2. What is the difference between the n-dimensional Riemann integral and other types of integrals?

The main difference between the n-dimensional Riemann integral and other types of integrals, such as the Lebesgue integral, is that it is defined in terms of partitions and Riemann sums. This means that it is based on dividing the region of interest into smaller subregions and approximating the function within each subregion, rather than using more advanced methods like measure theory.

3. How is the n-dimensional Riemann integral calculated?

The n-dimensional Riemann integral is calculated by dividing the region of interest into smaller subregions and approximating the function within each subregion. This is done using Riemann sums, which involve multiplying the function values by the size of each subregion and summing them together. As the size of the subregions decreases, the Riemann sums become more accurate and converge to the exact value of the integral.

4. What are some applications of the n-dimensional Riemann integral?

The n-dimensional Riemann integral has a wide range of applications in mathematics, physics, and engineering. It is used to calculate volumes and areas in higher dimensions, such as in multivariable calculus and differential geometry. It is also used in fields like computer graphics and image processing, where it can be used to calculate the volume of 3D objects or the area of complex surfaces.

5. Is there a limit to the number of dimensions that the Riemann integral can be applied to?

The Riemann integral can be applied to any number of dimensions, also known as the dimensionality of the space. However, as the number of dimensions increases, the calculations become more complex and may require advanced mathematical techniques. Additionally, the concept of volume or area may become more abstract in higher dimensions, making it difficult to visualize the results.

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