Q: Given that(adsbygoogle = window.adsbygoogle || []).push({}); G(x,y,z)=(6xz+x^{3}, 3x^{2}y+y^{2}, 4x+2yz-3z^{2}). FindFsuch that curlF=G.

Solution:

...

A particular solution is

Fo=(-3x^{2}yz-y^{2}z, 2x^{2}-3xz^{3}-x^{3}z)

And then my textbook says that the general solution is F=Fo + grad f where f is an arbitrary C^{1}function.

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Now my questions:

If f is C^{1}function, why must F=Fo+gradf be a solution to curl F=G?

I believe that curl(grad f)=0 for f a C^{2}(not C^{1}) function. Why does C^{1}work as well?

Secondly, why can we be sure that F=Fo+gradf is thegeneralsolution to curl F=G? (i.e. why iseverysolution contained in it?)

I would really appreciate if someone could explain.

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# Homework Help: Integrating Vector Derivatives

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