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Math Help - Gradient for Composite Functions

  1. #1
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    Gradient for Composite Functions

    I need to show that \nabla(au+bv)=a\nabla u + b\nabla v where a,b are constants and u=u(x,y),v=(x,y) are differentiale real-valued functions. If I have a function f(u,v)=au+bv where u,v are functions, then ultimately f(u,v)=f(x,y) right? Then I could right the i component of the gradient:

    \frac{\partial f}{\partial x}i=(\frac{\partial f}{\partial u}\frac{\partial u}{\partial x}+\frac{\partial f}{\partial v}\frac{\partial v}{\partial x})i

    Is my logic correct?
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  2. #2
    Junior Member slider142's Avatar
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    In the case of this theorem, think of the larger picture. The gradient is like a generalization of the single variable derivative. You must have already shown that the derivative is linear: d(af + bg) = a*df + b*dg, where a, b are constants and f and g are functions. Similarly, partial derivatives are also linear. Put these two facts together and you should be able to show that the gradient is linear without much work. As in most questions like these, go back to the definition. What does your book use as the definition of the gradient?
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