- #1
7thSon
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My apologies about lack of precision in nomenclature. So I wanted to know how to express a certain idea about choice of basis on a manifold...
Let's suppose I am solving a reaction-diffusion equation with finite elements. If I consider a surface that is constrained to lie in a flat plane or a volumetric solid in R3, the gradient operator and the diffusion tensor can be expressed in a Cartesian basis, and the gradient just reduces to partial derivatives.
It's come to my attention that if I am considering a non-planar surface embedded in R3, it is not valid to express the gradient operator in a Cartesian basis (i.e. d/dx, d/dy, d/dz); rather, you have to use covariant derivatives making use of surface coordinates.
My question is, what is the exact explanation as to why this is the case? Does it have something to do with the fact that the Cartesian basis will span all of R3, while the surface's tangent space is only a subset of R3? And, as such, the cartesian basis is an inappropriate choice of basis for the surface manifold?
Thanks for any help...
Let's suppose I am solving a reaction-diffusion equation with finite elements. If I consider a surface that is constrained to lie in a flat plane or a volumetric solid in R3, the gradient operator and the diffusion tensor can be expressed in a Cartesian basis, and the gradient just reduces to partial derivatives.
It's come to my attention that if I am considering a non-planar surface embedded in R3, it is not valid to express the gradient operator in a Cartesian basis (i.e. d/dx, d/dy, d/dz); rather, you have to use covariant derivatives making use of surface coordinates.
My question is, what is the exact explanation as to why this is the case? Does it have something to do with the fact that the Cartesian basis will span all of R3, while the surface's tangent space is only a subset of R3? And, as such, the cartesian basis is an inappropriate choice of basis for the surface manifold?
Thanks for any help...