Let F=(y^2e^4x+2xe^3y)i +G(x,y)j is a gradient field.

Find all G(x,y).

I'm confused all together.

Here's what I've done so far

I've found gradient function

4y^2e^4x+2e^3y+2ye^4x+6xe^3y

TBH, I don't understand what the question is asking..

Thanks

\(\displaystyle grad f(x,y) = \nabla f(x,y) = \frac{ \partial f }{ \partial x } \hat i + \frac{ \partial f }{ \partial y } \hat j \)

We are given \(\displaystyle \frac{ \partial f }{ \partial x } = y^2e^4x+2xe^3y \) so we need to find a Potential Function such that \(\displaystyle \frac{ \partial f }{ \partial y } = G(x,y) \)

To do this we will find F(x,y) by integration of the first part. Then find a constant such that our criteria is satisfied

\(\displaystyle \frac{ \partial f }{ \partial x } = y^2e^4x+2xe^3y \)

\(\displaystyle \partial f = y^2e^4x+2xe^3y \partial x \)

\(\displaystyle f(x,y) = \int y^2e^4x+2xe^3y dx = \frac{ y^2e^4x^2 }{2} + x^2e^3y + Q(y) \) where Q(y) is a function only dependant on y.

Therefore,

\(\displaystyle \frac{ \partial f }{ \partial y } = G(x,y) = ye^4x^2 + x^2e^3 + Q`(y) \)

To get to that step we simply differentiate our potential function with respect to y!