Geodesic
Simplified justification of the most important changes of variable in multiple integral
Matematičeskoe obrazovanie, Tome 95 (2020) no. 3, pp. 38-48

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The paper gives rigorous proofs of the transition in the integral to polar and spherical (including generalized) coordinates, significantly simplified compared to those available in the literature.
Keywords: change of variables in an integral, transition to polar coordinates, transition to spherical coordinates, general linear change in an m-fold integral, transformation formulas for the Jordan measure and multiple integral, the method of elementary Gaussian transformations. 
@article{MO_2020_95_3_a6,
     author = {S. S. Pukhov},
     title = {Simplified justification of the most important changes of variable in multiple integral},
     journal = {Matemati\v{c}eskoe obrazovanie},
     pages = {38--48},
     publisher = {mathdoc},
     volume = {95},
     number = {3},
     year = {2020},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MO_2020_95_3_a6/}
}
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S. S. Pukhov. Simplified justification of the most important changes of variable in multiple integral. Matematičeskoe obrazovanie, Tome 95 (2020) no. 3, pp. 38-48. http://geodesic.mathdoc.fr/item/MO_2020_95_3_a6/