Gradient of dot product using suffix notation

[spacetimedude]

Homework Statement

Find the gradient of [tex]\underline{\nabla}(\underline{a}\cdot\underline{r})^n[/tex] where a is a constant vector, using suffix notation and chain rule.

Homework Equations

On the previous problem,s I found that grad(a.r)=a and grad(r)=[itex]\underline{\hat{r}}[/itex]

The Attempt at a Solution

[tex]
\underline{e_i}(\frac{\partial }{\partial x_i})(\underline{a}\cdot\underline{r})^n=\underline{e_i}(\frac{\partial }{\partial x_i}(a_jx_j)^n)=\underline{e_i}(n(a_jx_j)^{n-1}(a_j\delta_{ij}))
[/tex]
I'm sure that the last step is wrong so could someone lead me to the right direction?
Thank you!

 

[HallsofIvy]

The derivative of [itex](ax)^n[/itex] is NOT [itex]n(ax)^{n-1}[/itex].

 

[spacetimedude]

The derivative of [itex](ax)^n[/itex] is NOT [itex]n(ax)^{n-1}[/itex].

That is what I thought. Could you explain that step?

 

[HallsofIvy]

Just after I wrote that I noticed that you also had "[itex]a_j\delta_{ij}[/itex]". The derivative of [itex](ax)^n= a^nx^n[/itex] is [itex]na^nx^{n-1}[/itex] but that can also be written as [itex](ax)^{n-2}(a)[/itex] where the last "a" is due to the chain rule. Was that what you meant?

 

[BvU]

##na^nx^{n-1}## is confusing. The inner product ##(a_j x_j)^{n-1}## is lost.

When I write it out and then put it back together again ##
\underline{e_i}(n(a_jx_j)^{n-1}(a_j\delta_{ij}))## seems OK.
##
na_i \underline{e_i}\;(a_jx_j)^{n-1}## might be somewhat more elegant,
and even that can be simplified further !