Short (I think) vector calculus question

[HmBe]

Homework Statement

Calculate the following expressions:

Homework Equations

The Attempt at a Solution

Letting the vector a = (a_1, a_2, a_3) I've worked out that it's 2|a|^2

While that method is fairly quick, I don't particularly like it, and was wondering if there is a shorter or neater method using identities and such, perhaps using del^2 = del . del, then using div and grad identities? Cheers.

 

[Dick]

You could work it out as div(grad((a.r)^2)), but it amounts to doing the pretty much the same thing you just did.

 

[HmBe]

Yeah, that's what I thought. But wasn't sure if there was a way to simplify grad((a.r)^2)

 

[Dick]

Yeah, that's what I thought. But wasn't sure if there was a way to simplify grad((a.r)^2)

grad((a.r)^2)=2(a.r)*grad(a.r)=2(a.r)a. div(2(a.r)a)=a.grad(2(a.r))+2(a.r)div(a)=2a.a+0=2a.a. You can write down all the steps with vectors, but I really don't think it's saving much time.