Einstein Notation - Not sure if my answers are right

[unscientific]

Homework Statement

Questions 11 and 12 specifically..

Homework Equations

The Attempt at a Solution

11(a)

11(b)

12(a)

12(b)

12(c)

12(d)

(I did the last part of 12(d) by normal vector methods and got 2a instead...which is the correct answer)

 

[vela]

An index shouldn't appear more than twice. For the first one, for example, you should have
$$\frac{\partial}{\partial r_i} (r_j r_j)^{n/2}$$

 

[unscientific]

An index shouldn't appear more than twice. For the first one, for example, you should have
$$\frac{\partial}{\partial r_i} (r_j r_j)^{n/2}$$

Thanks for your reply!

Also, I can't see what's wrong with my final answer in part 12(d)...i think the last line is the cross product of ∂_j and r which equates to zero??

I think I have figured out every question other than part 12(d)

 

[vela]

Hint: $\partial_i r_j = \delta_{ij}$. You can use that in several of the problems. In 12b, you need to use the product rule.

 

[unscientific]

Hint: $\partial_i r_j = \delta_{ij}$.

thanks i think I've sorted it out!

 

[WannabeNewton]

what does this mean? is it a vector or a scalar? summation implied or not?

It is not a summation. It is just the statement that the partial of the jth coordinate function with respect to the ith coordinate function is just the kronecker delta.