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- TL;DR Summary
- By gradient theorem, if f(x)=grad(F(x)), where F(x) is conservative field, then closed line integral of f is zero. Books tell me that also g(F(x)) * grad(F(x)) is zero. I can't derive why.

This problem comes from fluid dynamics where Kelvin circulation theorem states, that if density "rho" is a function of only pressure "p", then closed line integral of grad(p) / rho(p) equals zero. It seems so trivial, so that no one ever gives reason for this claim.

When trying to solve it, I've generalized it to the following:

If f(x)=grad(F(x)), then closed line integral of f(x) is zero, by gradient theorem. From this, it somehow follows that g(F(x)) * grad(F(x)) is also zero.

I tried to rewrite:

g(F(x)) * grad(F(x)) = grad(g(F(x)) * F(x)) - grad(g(F(x))) * F(x), by a gradient identity similar to derivative of product

And somehow show that the expanded form can be rewritten as a gradient of some function. I did not succeed.

Can somebody help me?

Thank you

When trying to solve it, I've generalized it to the following:

If f(x)=grad(F(x)), then closed line integral of f(x) is zero, by gradient theorem. From this, it somehow follows that g(F(x)) * grad(F(x)) is also zero.

I tried to rewrite:

g(F(x)) * grad(F(x)) = grad(g(F(x)) * F(x)) - grad(g(F(x))) * F(x), by a gradient identity similar to derivative of product

And somehow show that the expanded form can be rewritten as a gradient of some function. I did not succeed.

Can somebody help me?

Thank you