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.

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

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.

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Integrating Vector Derivatives

**Physics Forums | Science Articles, Homework Help, Discussion**