Divergence of vector field: Del operator/nabla

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Homework Help Overview

The discussion revolves around the divergence of a vector field defined as ν(x,y,z) = (xi + yj + zk)rk, where r is the magnitude of the position vector. Participants are tasked with showing that ∇.v = λrk except at r=0 and finding λ in terms of k.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning, Assumption checking

Approaches and Questions Raised

  • Participants explore the definition of the del operator and the dot product, questioning the application of these concepts in the context of the problem.
  • There are attempts to compute the divergence and discussions about the correct application of the product and chain rules during differentiation.
  • Some participants express confusion about the manipulation of the divergence expression and the interpretation of the variable k.
  • Questions arise regarding the dependence of r on x, y, and z, and how that affects the differentiation process.

Discussion Status

The discussion is active, with participants providing insights and corrections to each other's reasoning. Some have suggested viable approaches to compute the divergence, while others are still clarifying their understanding of the underlying concepts. There is no explicit consensus yet, but the dialogue is productive.

Contextual Notes

Participants note the complexity introduced by the dependence of r on the Cartesian coordinates and the need to apply the product and chain rules correctly. There are also references to external resources that may influence the understanding of the problem.

  • #31
It's very close to it though! I'm pretty happy, I understand that so much better.
The derivatives are right according to wolfram. So that's one thing ruled out.
 
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  • #32
Aha! It is in the derivative (rather the substitution). See if you can find it.
So what I actually had was
##x\frac{k}{2}u^{\frac{k}{2}-1}*2x= x^2*u^{\frac{k}{2}-1}##
Subbing back in for r (r = u^1/2) that should be ##x^2r^{k-2}## rather than k-1
 
  • #33
BiGyElLoWhAt said:
Aha! It is in the derivative (rather the substitution). See if you can find it.
So what I actually had was
##x\frac{k}{2}u^{\frac{k}{2}-1}*2x= x^2*u^{\frac{k}{2}-1}##
Subbing back in for r (r = u^1/2) that should be ##x^2r^{k-2}## rather than k-1
I used the chain rule on (x2+y2+z2)0.5k though, and taking this approach I don't see where a k-2 comes from...
 
  • #34
the chain rule is actually being applied twice. If you see my chain rule that I posted earlier, you ultimitely need to take the derivative with respect to u. that's where that expression comes from. So what you have there is fine, but that's u^k/2, not r. You need to substitute that back in. So you have ##\frac{k}{2}u^{0.5k - 1}## with me so far?
this is equal to ##\frac{k}{2}\frac{u^{.5k}}{u}## you need to use the fact that u = r^2 (because r has the square root in it)
##u^.5k = r^k## and ##1/u = r^{-2}## so all together it's ##\frac{k}{2}r^{k-2}## I might have lost some constants along the way, I'm kinda in a hurry atm, walking out the door at work, but that's where the 2 comes in, and that's what gets rid of the r in the lambda.
 
  • #35
Oh, got it! And that leads to the 3+k. Phew! What a question. I really, really appreciate all your time and help, thank you :)
 
  • #36
Hey not a problem, that's what we're here for.
 
  • #37
who's serving things up on a silver platter now? :wink:
 
  • #38
XD
 
  • #39
Eh, we got to "the answer", it was just wrong. Substitution errors and whatnot haha.
 

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