Basis + covariant derivative question

[7thSon]

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...

 

[tiny-tim]

Hi 7thSon!

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?

Using the covariant derivative makes the (partial) derivative of each coordinate unit vector zero …

it would be really inconvenient not to have ∂x₂/∂x₁ = 0 etc …

for example, div of ai + bj would not be ∂a/∂x + ∂b/∂y.

 

[7thSon]

Hey thanks for the reply. I get what you're saying, but I guess I wasn't being clear about the exact cause of my confusion.

So if I understand you correctly, I believe that yes, with a Cartesian basis the base vectors are not changing, so when taking the covariant derivative at that tangent space it reduces to partial derivatives because the derivative of the basis vectors is zero.

However, what I don't get is why for a general curved surface, if I write the gradient operator with the Cartesian base vectors and solve the correct linear system, it gives you the wrong answer for a diffusion on a surface, whereas the covariant derivative in surface coordinates gives you the right answer.

Why is the former approach wrong? The answer must have to do something with the choice of base vectors (E3) being inappropriate for solving a conservation law strictly on a surface.