Generalized Divergence Theorem and Covariant Derivative
Explore the concepts of the Generalized Divergence Theorem and Covariant Derivative through visual representations and explanations in index notation, with insights on surface derivatives, change in surface basis vectors, and more. Delve into advanced mathematical theories in this informative collection of images.
Download Presentation
Please find below an Image/Link to download the presentation.
The content on the website is provided AS IS for your information and personal use only. It may not be sold, licensed, or shared on other websites without obtaining consent from the author. If you encounter any issues during the download, it is possible that the publisher has removed the file from their server.
You are allowed to download the files provided on this website for personal or commercial use, subject to the condition that they are used lawfully. All files are the property of their respective owners.
The content on the website is provided AS IS for your information and personal use only. It may not be sold, licensed, or shared on other websites without obtaining consent from the author.
E N D
Presentation Transcript
Generalized Divergence Theorem Divergence theorem ? ?? = ? ??? ? ?
Generalized Divergence Theorem Divergence theorem Index notation ????? = ?????? ? ?? = ? ??? ? ? ? ?
Generalized Divergence Theorem Divergence theorem Index notation ????? = ?????? ? ?? = ? ??? ? ? ? ? ????? = ?????? ? ?
Another proof, in index notation Definition of the Covariant Derivative ??? ???= ????+ ?? ???= 0 ???= ???? ?? ???= ???? ??
Another proof, in index notation Definition of the Covariant Derivative Generalized Divergence Theorem ??? ???= ????+ ?? ????? = ?????? ???= 0 ???= ???? ?? ? ? ???= ???? ??
T?? ??????? ????? ?? ?/??? ? a ? ? b ?/??? ? + ?? ? ?/??? ? + ?? ? c ? d ? ? + ?? ? 2 ???? ? + 1 + ?? ? + ?? ?
T?? ??????? ????? ?? ??? ???? ???= ???? ?? ?? but
T?? ??????? ????? ?? ??? ???? ???= ???? ?? ?? but instead ?= ?? ???? ?= ?? ???? ?? ?? and
T?? ??????? ????? ?? ??? ???? ???= ???? ?? ?? but instead ?= ?? ???? ?= ?? ???? ?? ?? and You still have a covariant surface derivative ? but ??? 0, instead: See Grinfeld 11.5 and 12.4
Change in Surface Basis Vectors ?? ?(?) ? ?(?)
Change in Surface Basis Vectors ?? ??(?) =?? ??? ?(?) ? ?(?)
Change in Surface Basis Vectors ?? ??(?) =?? ??? ?(?) ?(? + ) ?? ???+ ??(? + ) = ? ?(?) ?(? + )
Change in Surface Basis Vectors ?? ??(?) =?? ??? ?(?) ??? + ??(?) ?(? + ) ?? ???+ ??(? + ) = ? ?(?) ?(? + )
Change in Surface Basis Vectors ?? ??(?) =?? ??? ?(?) ??? + ??(?) ?(? + ) ?? ???+ ??(? + ) = ? ??? ??= lim ??? + ??(?) 0 ?(?) ?(? + )
Failure on a curved surface ??? 0 implies that in general:
Failure on a curved surface ??? 0 implies that in general: Though on flat surfaces, we still have Where the boxed equality still holds in general by the Divergence Theorem
What does it mean to generalize? Divergence theorem ? ?? = ? ??? ? ? Gradient tensor theorem Generalized Stokes theorem ?? = ? ? ??
References Grinfeld, Pavel. Introduction to tensor analysis and the calculus of moving surfaces. New York: Springer, 2013.
