How Do You Use Einstein Summation to Prove Vector Calculus Identities?

[mmpstudent]

prove the identity $$\nabla\times(f\cdot\vec{v})=(\nabla f) \times \vec{v} + f \cdot \nabla \times \vec{v}$$

I can do the proof with normal vector calculus, but I am in a tensor intensive course and would like to do this with
einstein summation notation, but am having some trouble since I am brand new to this.

my attempt

left side

$$\epsilon_{ijk} \partial_{j} (\nabla f \cdot \vec{v})_{k}=\epsilon_{ijk}\partial_{j} f \vec{v}_{k}$$

I didn't really know where to go from here so I moved onto the right side and expressed it in einstein notation

$$\epsilon_{ijk} (\nabla f)_{j} v_{k} + f \epsilon_{ijk} \partial_{j} v_{k}$$

$$\epsilon_{ijk} \partial_{j} f v_{k} + f \epsilon_{ijk} \partial_{j} v_{k}$$

which I don't see how I can rearrange this to get what is on the left. I see how it could be twice what I have on the left, but that obviously is incorrect. Did I do something wrong in expressing these? Do I have to express the right side with different sets of indicees?

 

[George Jones]

How did you go from the second last line to the last line?

 

[mmpstudent]

I just realized I forgot an f in that line

 

[George Jones]

If you include the f, haven't you answered your own question?

 

[mmpstudent]

If you include the f, haven't you answered your own question?

unless I am seeing this completely wrong, the left side (first line up above) and the right side (on the last line) is twice the left side when I add them together

I was thinking maybe I had to express the right side like this

$$\epsilon_{ijk} (\nabla f)_{j} v_{k} + f \epsilon_{klm} \partial_{l} v_{m}$$

and do the permutation identity for permutations differing by 2 indicees but I seem to be going nowhere with that

 

[George Jones]

Apply the product rule to

$$\epsilon_{ijk}\partial_{j} (f v_{k})$$

 

[mmpstudent]

Apply the product rule to

$$\epsilon_{ijk}\partial_{j} (f v_{k})$$

jeez thanks... staring me in the face

 

[mmpstudent]

Can anyone suggest a book that has a ton of examples using einstein summation? I feel behind most of my class in regards to the notation. It just takes me too long to do problems.

 

[Fredrik]

Just a small tip: Don't use $\cdot$ for anything other than the dot product when you're doing these things.