- #1

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**G**(x,y,z)=(6xz+x

^{3}, 3x

^{2}y+y

^{2}, 4x+2yz-3z

^{2}). Find

**F**such that curl

**F**=

**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.

===============================

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 the

*general*solution to curl F=G? (i.e. why is

*every*solution contained in it?)

I would really appreciate if someone could explain.