Curl of a gradient and the anti Curl

  • #1
LaplacianHarmonic
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Homework Statement


Is there a vector field D that produces The position vector <x,y,z> if we take the curl of vector field D?

Homework Equations


Curl of gradient f = 0

Curl of Vector D = <x,y,z>


The Attempt at a Solution



Curl of vector D
Where vector D=<A,B,C>

Cy - Bz = x
Az - Cx = y
Bx - Ay = z

I can't solve what component functions A, B, C are.

HELP[/B]
 

Answers and Replies

  • #2
Fred Wright
362
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You are asked to determine if a vector exists s.t. $$ \vec \nabla \times \vec D = \vec r$$
I suggest you expand the cross product in it's components. For example, the x component would be:$$\partial_y D_z -\partial_z D_y=x$$
Clearly ##D_z## must be of the form ##a_zyx##, where ##a_z## is a constant. Do the same for the other two components.
 
Last edited:
  • #3
LCKurtz
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Homework Statement


Is there a vector field D that produces The position vector <x,y,z> if we take the curl of vector field D?
Think about what you know about the divergence of a curl...
 
  • #4
LaplacianHarmonic
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Divergence of a curl is zero
 
  • #6
LaplacianHarmonic
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So... divergence of a curl measures how much the vector diverges outward after measuring how much that vector was curling. Thus, it is always zero.
 
  • #7
LCKurtz
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But what does that observation have to do with your problem?
 

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