# Vector calculus index notation

Physgeek64

## Homework Equations

curl(a x b)= (b dot grad)a - (a dot grad)b +a(div b) - b(div a )

## The Attempt at a Solution

Im trying to use index notation and get

which is obviously not right. Ive tried attacking the problem from the reverse direction and haven't had much luck there either.

Thank you :)

Mentor
Can you provide more details? Is A a constant vector not dependent on r, theta, phi?

Have you looked at the vector identities in spherical coordinates for curl, div and grad?

Try to solve it without going to index notation.

Mentor
Try starting with ##\nabla( A \cdot B ) = (B \cdot \nabla ) A + (A \cdot \nabla) B + B \times (\nabla \times A) + A \times (\nabla \times B)##

Physgeek64
Try starting with ##\nabla( A \cdot B ) = (B \cdot \nabla ) A + (A \cdot \nabla) B + B \times (\nabla \times A) + A \times (\nabla \times B)##
Sorry, a is a constant vector and r is (X,y,z)

I have not come across this identity. Any hints on how to prove it?

Mentor
Here's a more complete list of identities:

https://en.wikipedia.org/wiki/Vector_calculus_identities

Off hand I don't know how to prove this particular identity but often when presented with a problem such as yours you can use the identities transform the left had side to the right hand side or vice versa.to "prove" it.

Homework Helper
Gold Member
2021 Award

## Homework Equations

curl(a x b)= (b dot grad)a - (a dot grad)b +a(div b) - b(div a )

## The Attempt at a Solution

Im trying to use index notation and get

which is obviously not right. Ive tried attacking the problem from the reverse direction and haven't had much luck there either.

Thank you :)

Time to learn some latex, I suggest!

Mentor
Some latex codes:

http://web.ift.uib.no/Teori/KURS/WRK/TeX/symALL.html

in particular look at vector operators \nabla for the del operator,\cdot for the inner-product and \times for the cross-product. This site uses mathjax for display. Mathjax looks for expressions bracketted by double # characters.

Physgeek64
Here's a more complete list of identities:

https://en.wikipedia.org/wiki/Vector_calculus_identities

Off hand I don't know how to prove this particular identity but often when presented with a problem such as yours you can use the identities transform the left had side to the right hand side or vice versa.to "prove" it.
I think proving the identity was the purpose of the question? I could be wrong but it's to do with index notation

Mentor
You problem has a specific function ##\nabla(1/r)## in it so its seemed to me that its not an identity but an example where you apply the identity to get the result you need to prove.

Its true I could be wrong here. Perhaps @Mark44 or @PeroK could comment more on this.

Homework Helper
Gold Member
2021 Award
You problem has a specific function ##\nabla(1/r)## in it so its seemed to me that its not an identity but an example where you apply the identity to get the result you need to prove.

Its true I could be wrong here. Perhaps @Mark44 or @PeroK could comment more on this.

I assumed it did depend on the second vector being ##\nabla(1/r)##. It must. I evaluated each side for the x-term (and then used symmetry). It's not too bad if you're careful with your differentiation.

jedishrfu
Homework Helper
Gold Member
2021 Award
I assumed it did depend on the second vector being ##\nabla(1/r)##. It must. I evaluated each side for the x-term (and then used symmetry). It's not too bad if you're careful with your differentiation.

I checked with the second vector being ##\nabla f## and it reduces to Laplace's equation ##\nabla ^2 f = 0##

jedishrfu
Physgeek64
I checked with the second vector being ##\nabla f## and it reduces to Laplace's equation ##\nabla ^2 f = 0##
Its okay, Ive managed to do it with index notation. Thank you guys for helping though- I really appreciate it :)