Proving Vector Calculus Identities: Tips and Tricks

[bothcats]

Homework Statement

div(øu) = ødivu + ugradø

Homework Equations

divergence of scalar field = f,_ii
divergence of vector field = u_i,_i

The Attempt at a Solution

I've heard this is a simple proof, but this is my first one of 8 or so proofs I need to complete for homework, and I'm really not sure where to start. I know that div v = ∇ . v, but that's as far as I've gotten. We need to use Index Notation. Thoughts on where to start?

 

[Fredrik]

Homework Statement

div(øu) = ødivu + ugradø

Homework Equations

divergence of scalar field = f,_ii
divergence of vector field = u_i,_i

The Attempt at a Solution

I've heard this is a simple proof, but this is my first one of 8 or so proofs I need to complete for homework, and I'm really not sure where to start. I know that div v = ∇ . v, but that's as far as I've gotten. We need to use Index Notation. Thoughts on where to start?

Why not start by applying the definition of the divergence of a vector field to $\phi\mathbf u$? Please don't use the empty set symbol instead of $\phi$. I found it very confusing, and it took me some time to understand what you meant. If you don't use LaTeX, and can't find another way to type a $\phi$, then just call the scalar field f or something like that.

 

[bothcats]

Sorry for the confusion. I'll be more careful with the lettering in the future. I've actually figured this one out now. It was the product rule that I wasn't sure about, now that I've worked it through (and several other identity proofs). Now, I'm on the divergence of (u cross v) identity.

Thanks!