Short (I think) vector calculus question

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Homework Help Overview

The discussion revolves around a vector calculus problem involving the calculation of certain expressions, specifically focusing on the use of vector identities and operations such as divergence and gradient.

Discussion Character

  • Exploratory, Mathematical reasoning, Assumption checking

Approaches and Questions Raised

  • Participants discuss various methods to calculate the expressions, including the original poster's approach using the magnitude of a vector. There is a consideration of whether alternative methods, such as using divergence and gradient identities, could yield a more efficient solution.

Discussion Status

The conversation is ongoing, with participants exploring different methods and questioning the efficiency of their approaches. Some guidance has been offered regarding the use of divergence and gradient, but no consensus has been reached on a preferred method.

Contextual Notes

Participants are considering the implications of using vector identities and the potential for simplification in their calculations. There is an acknowledgment of the complexity involved in the expressions being discussed.

HmBe
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Homework Statement



Calculate the following expressions:

question.png


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

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