The divergence theorem for vector-valued forms

• Autorzy: Pierzchalski Antoni

Identyfikatory

DOI

10.17512/jamcm.2024.4.08

• Języki publikacji: EN

Abstrakty

EN

The divergence theorem for a vector valued form of any degree p = 0,1, . . . ,n is derived on a Riemannian manifold M of dimension n with a nonempty boundary ∂M. In analogy to the classic theorem, it relates the integration over M to the integration over ∂M. In the particular case p = 0, when the vector valued form reduces to a vector field, the theorem reduces to the classic divergence theorem.

Słowa kluczowe

EN

vector valued forms; the gradient; the divergence; the divergence theorem

PL

formy wektorowe; gradient; dywergencja; twierdzenie o dywergencji

• Wydawca: Wydawnictwo Politechniki Częstochowskiej
• Czasopismo: Journal of Applied Mathematics and Computational Mechanics
• Rocznik: 2024
• Tom: Vol. 23, nr 4
• Strony: 89--100
• Opis fizyczny: Bibliogr. 10 poz.

Twórcy

autor

Pierzchalski Antoni

• Faculty of Mathematics and Computer Science, University of Lodz , Poland

Bibliografia

• 1. Spivak, M. (2018). Calculus on Manifolds, a Modern Approach to Classical Theorems of Advanced Calculus. CRC Press.
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• 3. Nakahara, M. (2003). Geometry, Topology and Physics. Institute of Physics Publishing.
• 4. De, D. (2022). Introduction to Differential Geometry with Tensor Applications. Modern Mathematics in Computer Science Wiley-Scrivener.
• 5. Nguyen-Schaffer, H., & Schmidt, J.-P. (2017). Tensor Analysis and Elementary Differential Geometry for Physicists and Engineers. Mathematical Engineering, Springer.
• 6. Eifler, L., & Rhee, N. (2008). The n-dimensional Pythagorean theorem via the divergence theorem. Amer. Math. Monthly, 115, 5, 456-457.
• 7. Rummler, H. (1989). Differential forms, Weitzenb¨ock formulae and foliations. Publications Mathematiques, 33, 543-554.
• 8. Klekot, A, & Pierzchalski, A. (2023). The garadients and the divergence for vector valued forms. J. Appl. Math. Comput. Mech., 22, 1, 27-41.
• 9. Kozłowski, W., & Pierzchalski, A. (2008). Natural boundary value problems for weighted form Laplacians. Ann. Scuola Norm. Sup. Pisa Cl. Sci., 7(5), 343-367.
• 10. Pierzchalski A. (2017). Gradients: the ellipticity and the elliptic boundary conditions – a jigsaw puzzle. Folia Mathematica, 19, 65-83.

Uwagi

Opracowanie rekordu ze środków MNiSW, umowa nr POPUL/SP/0154/2024/02 w ramach programu "Społeczna odpowiedzialność nauki II" - moduł: Popularyzacja nauki (2025).