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Short (I think) vector calculus question

  • Thread starter HmBe
  • Start date
  • #1
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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.
 

Answers and Replies

  • #2
Dick
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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.
 
  • #3
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Yeah, that's what I thought. But wasn't sure if there was a way to simplify grad((a.r)^2)
 
  • #4
Dick
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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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