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Can you find the gradient of a vector?

  1. Jan 16, 2013 #1
    1. The problem statement, all variables and given/known data

    I know you can find the gradient of a scalar using partial derivatives. Does it make sense to find the gradient of a vector, however?

    A homework problem of mine asks to find the gradient of a vector. I'm starting to think it's a trick question....

    2. Relevant equations

    ∇ dot V = the divergence of V
    ∇ cross V = the curl of V

    3. The attempt at a solution

    The equations above lead me believe that it doesn't make sense to take the gradient of a vector , but the gradient operator can be used in combination with a dot product or cross product to give similar information about the way a function behaves (divergence and curl). So, perhaps divergence and curl are like the vector version of a gradient?
     
  2. jcsd
  3. Jan 16, 2013 #2
    V∇ !
    Its not a trick question!
     
    Last edited: Jan 16, 2013
  4. Jan 16, 2013 #3
    Sorry, my above comment makes no sense, that was the gradient of a scalar which you mentioned, I read too fast! I'm curious too....maybe someone more math minded can help?...
     
    Last edited: Jan 16, 2013
  5. Jan 16, 2013 #4

    Dick

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    You can make the gradient of vector make some sense if you know what a tensor is. If not, then you may have misinterpreted the question. What is it?
     
  6. Jan 17, 2013 #5
    The specific question is "You are given some vector function V(x,y,z). Can the gradient operation be operated on V? If so, how would you interpret the result?" Very vague, I know.

    I have not yet learned what a tensor is. My teacher is definitely one to give tricky questions, so I'm sort of just thinking this is a way to teach us that the gradient operation is used on scalars, and the divergence and curl operations are used on vectors. Is this correct? The way I make sense of this in my head is with an analogy like divergence & curl : vector as gradient: scalar. What do you think?
     
  7. Jan 17, 2013 #6

    Dick

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    Ok, so it's just a thinking question. Gradient gives you a vector from a scalar. If you want to take the gradient of a vector you might think about taking the gradient of each component giving you a matrix.
     
  8. Jan 17, 2013 #7

    Ray Vickson

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    One way to think about the gradient is as a "linearization factor", so if v(x,y,z) is a scalar function we have
    [tex] v(x+h_x, y+h_y z+h_z) = v(x,y,z) + <A,h> + O(|h|^2), [/tex] where A is a vector, h = (h_x,h_y,h_z) is a vector and <.,.> denotes the inner product. If that holds for all h, we must have A = grad v. Can you think of a similar representation when v is a vector?
     
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