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math_proof

  • 2 years ago

finding potential function for F=<-y,-x> is it just C? or is it -yx-xy+C?

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  1. zepdrix
    • 2 years ago
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    \[\huge \vec{F}=-y\hat{i}-x\hat{j}\] \[\large \vec{F}=M\hat{i}+N\hat{j}\]\[\large M=\frac{\partial f}{\partial x} \qquad N=\frac{\partial f}{\partial y}\]We need to make sure that the second partials of M and N match,\[\large M_y=-1 \qquad N_x=-1\]Looks good. Soooo, hmm.. we'll integrate M with respect to x giving us, \[\large -yx+g(y)\]If we integrate N with respect to y we get, \[\large −xy+h(x)\] So we need to look at each equation, see what part accounts for g(y) and h(x). Since -xy appears in both pieces, there is nothing else missing. The other term is simply a constant.\[\huge f(x,y)=-xy+c\] I don't have the best understanding of this stuff quite yet, but hopefully that helps D: Lemme know if any of it is confusing.

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