- #1

Peregrine

- 22

- 0

I am playing around with learning index notation for tensors, and I came across the following where C is a 0th order tensor:

[tex]E_{ijk} \partial_j \partial_k C = 0[/tex]

I believe this equates to [tex]\nabla \times \nabla C[/tex]. I don't understand why this comes out to 0. Any ideas?

Also, I am trying to understand in index notation how to represent the grad of a vector. The reason I am confused is that it seems that, taking C as a 0th order tensor, V as a 1st order tensor and T as a 2nd order tensor:

[tex]div T = \nabla \cdot T = \partial_iT_{ij}[/tex]

[tex]div V = \nabla \cdot V = \partial_iV_i[/tex]

And of course, div C does not make sense as it would be a -1st order tensor.

But, since:

[tex]grad C = \nabla C = \partial_iC[/tex]

I don't follow how to represent [tex]grad V = \nabla V[/tex] or [tex]grad T = \nabla T[/tex] in index notation; from what I have it seems there would be no difference in notation between grad and div! Any help would be greatly appreciated. Thanks!

[tex]E_{ijk} \partial_j \partial_k C = 0[/tex]

I believe this equates to [tex]\nabla \times \nabla C[/tex]. I don't understand why this comes out to 0. Any ideas?

Also, I am trying to understand in index notation how to represent the grad of a vector. The reason I am confused is that it seems that, taking C as a 0th order tensor, V as a 1st order tensor and T as a 2nd order tensor:

[tex]div T = \nabla \cdot T = \partial_iT_{ij}[/tex]

[tex]div V = \nabla \cdot V = \partial_iV_i[/tex]

And of course, div C does not make sense as it would be a -1st order tensor.

But, since:

[tex]grad C = \nabla C = \partial_iC[/tex]

I don't follow how to represent [tex]grad V = \nabla V[/tex] or [tex]grad T = \nabla T[/tex] in index notation; from what I have it seems there would be no difference in notation between grad and div! Any help would be greatly appreciated. Thanks!

Last edited: