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Math Help - Gradient

  1. #1
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    Gradient

    The gradient is just the jacobian matrix correct

    f(x,y) = \log \sqrt{x^2 + y^2}

    \nabla f(x,y) = \left[\begin{array}{cc} \frac{1}{\ln 10}\frac{x}{x^2+y^2}&\frac{1}{\ln 10} \frac{y}{x^2 +y^2} \end{array}\right] ??

    also, I went to a couple differential geometry talks before and they were referring to nabla notation, were they referring to the gradient?
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  2. #2
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    Re: Gradient

    The gradient of a function of two variables is the vector \frac{\partial f}{\partial x}\vec{i}+ \frac{\partial f}{\partial y}\vec{j}. I'm not sure I would call that the "Jacobian matrix". The "nabla" is the \nabla operator which can be thought of a kind of "symbolic vector": \frac{\partial}{\partial x}\vec{i}+ \frac{\partial}{\partial y}\vec{j} or, in three dimensions \frac{\partial}{\partial x}\vec{i}+ \frac{\partial}{\partial y}\vec{j}+ \frac{\partial}{\partial z}.

    The gradient can be thought of a product of that "vector" with a scalar function: \nabla f= (\frac{\partial}{\partial x}\vec{i}+ \frac{\partial}{\partial y}\vec{j}+ \frac{\partial}{\partial z})f = \frac{\partial f}{\partial x}\vec{i}+ \frac{\partial f}{\partial y}\vec{j}+ \frac{\partial f}{\partial z}\vec{k}.

    But you can also take the dot product of the "vector" operator and a vector valued function: \nabla\cdot \vec{F}= (\frac{\partial}{\partial x}\vec{i}+ \frac{\partial}{\partial y}\vec{j}+ \frac{\partial}{\partial z}\vec{k})\cdot (f(x,y,z)\vec{i}+ g(x,y,z)\vec{j}+ h(x,y,z)\vec{k}) = \frac{\partial f}{\partial x}+ \frac{\partial g}{\partial y}+ \frac{\partial h}{\partial z}. That is the "divergence of the vector valued function" or div(\vec{F}).

    Or you can take the cross product of the "vector" operator and a vector valued function: \nabla\times \vec{F}= ((\frac{\partial}{\partial x}\vec{i}+ \frac{\partial}{\partial y}\vec{j}+ \frac{\partial}{\partial z}\vec{k})\times (f(x,y,z)\vec{i}+ g(x,y,z)\vec{j}+ h(x,y,z)\vec{k})= \left(\frac{\partial h}{\partial y}- \frac{\partial g}{\partial z}\right)\vec{i}+ \left(\frac{\partial f}{\partial z}- \frac{\partial h}{\partial x}\right)\vec{j}+ \left(\frac{\partial g}{\partial x}- \frac{\partial f}{\partial y}\right)\vec{k}.
    That is the "curl of the vector valued function" or curl(F).
    Last edited by HallsofIvy; July 4th 2014 at 05:54 PM.
    Thanks from Jonroberts74
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  3. #3
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    Re: Gradient

    okay cool, I can see the similarity between derivative(jacobian) matrix and gradient.

    thanks
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