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Math Help - Flow Lines/Curl Question?

  1. #1
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    Flow Lines/Curl Question?

    The curl of the gradient of a twice continuously differentiable function R3 -> R is identically zero.
    Prove this by direct computation of the required mixed partial derivatives.
    Last edited by Playthious; December 2nd 2010 at 03:04 PM.
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  2. #2
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    I am not 100% on what you are asking. Is there more to this question? However, I have devised a guess which is provided below.

    \displaystyle \begin{vmatrix}<br />
\mathbf{i} & \mathbf{j} & \mathbf{k}\\ <br />
\frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z}\\ <br />
F_x & F_y & F_z<br />
\end{vmatrix}=\left(\frac{\partial z}{\partial y}-\frac{\partial y}{\partial z}\right)\mathbf{i}+\left(\frac{\partial x}{\partial z}-\frac{\partial z}{\partial x}\right)\mathbf{j}+\left(\frac{\partial y}{\partial x}-\frac{\partial x}{\partial y}\right)\mathbf{k}
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  3. #3
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    there was a second part that said what can you conclude about the effect on local circulation of grad f?

    I'm not sure how to find the mixed partial derivatives though
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  4. #4
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    The ones I posted?
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  5. #5
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    yea because it never actually mentions any function
    it just says "a twice differentiable function"
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  6. #6
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    If you take the determinant of the matrix (cross product), you will obtain those partial derivatives.
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