Abstract
In this chapter, we shall obtain important relations between different types of integrals in 3D. The generalizations to an arbitrary dimension are beyond the scope of this book since the proofs would require too much space. One can find these general versions of Gauss and Stokes theorems in other books, e.g., in [1] or [2].
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Notes
- 1.
As an example we may take internal coordinates \(\,\varphi ,\theta \,\) on the sphere. We can introduce one more coordinate \(\,r\,\) and the new coordinates \(\,\varphi ,\,\theta ,\,r\,\) will of course be the spherical ones, which cover all the 3D space. The only one difference is that the new coordinate r, in this example, is not zero on the surface \(r=R\). But this can be easily corrected by the redefinition \(\rho =r-R\), where \(\,-R<\rho <\infty \).
References
A.T. Fomenko, S.P. Novikov, B.A. Dubrovin, Modern Geometry-Methods and Applications, Part I: The Geometry of Surfaces, Transformation Groups, and Fields (Springer, Berlin, 1992)
W. Rudin, Principles of Mathematical Analysis, 3 edn. (McGraw-Hill, New York, 1976)
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Shapiro, I.L. (2019). Theorems of Green, Stokes, and Gauss. In: A Primer in Tensor Analysis and Relativity. Undergraduate Lecture Notes in Physics. Springer, Cham. https://doi.org/10.1007/978-3-030-26895-4_9
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