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Ok so I'm new to vector analysis, just started about a week or 2 ago. I'm using Paul C. Matthews' book, "Vector Calculus". This is an example problem from it which I have difficulty understanding because of integration with partial derivatives. The problem is solved, I just have trouble understanding the solution.

Show that the vector field [tex]F = (2x+y, x, 2z)[/tex] is conservative.

So if [tex]F[/tex] is conservative, it can be written as the gradient of some scalar field [tex]\phi[/tex]. This gives the three equations:

[tex]\frac{\partial \phi}{\partial x}[/tex], [tex]\frac{\partial \phi}{\partial y}[/tex] and [tex]\frac{\partial \phi}{\partial z}[/tex]

After this, they integrate first of the equations with respect to x and this gives [tex]\phi = x^2+xy+h(x,y)[/tex]. This is still alright, because h is analogous to the constant of integration. But after this it says:

"The second equation forces the partial derivative of h with respect to y to be zero so that h only depends on z. The third equation yields [tex]\frac{dh}{dz}=2z[/tex] so [tex]h(z) = z^2+c[/tex] where c is any constant. Therefore, all three equations are satisfied by the potential function [tex]\phi = x^2+xy+z^2[/tex] and F is a conservative vector field"

I didn't really understand much of what is in the quotes except this part:

[tex]\frac{\partial \phi}{\partial y} = x[/tex] and [tex]\frac{\partial (x^2+xy+h(x,y))}{\partial y} = x+\frac{\partial h(x,y)}{\partial y}[/tex] and so h(x,y) is constant but why does this mean that h only depends on z?

Thanks if you can help.

## Homework Statement

Show that the vector field [tex]F = (2x+y, x, 2z)[/tex] is conservative.

So if [tex]F[/tex] is conservative, it can be written as the gradient of some scalar field [tex]\phi[/tex]. This gives the three equations:

[tex]\frac{\partial \phi}{\partial x}[/tex], [tex]\frac{\partial \phi}{\partial y}[/tex] and [tex]\frac{\partial \phi}{\partial z}[/tex]

After this, they integrate first of the equations with respect to x and this gives [tex]\phi = x^2+xy+h(x,y)[/tex]. This is still alright, because h is analogous to the constant of integration. But after this it says:

"The second equation forces the partial derivative of h with respect to y to be zero so that h only depends on z. The third equation yields [tex]\frac{dh}{dz}=2z[/tex] so [tex]h(z) = z^2+c[/tex] where c is any constant. Therefore, all three equations are satisfied by the potential function [tex]\phi = x^2+xy+z^2[/tex] and F is a conservative vector field"

I didn't really understand much of what is in the quotes except this part:

[tex]\frac{\partial \phi}{\partial y} = x[/tex] and [tex]\frac{\partial (x^2+xy+h(x,y))}{\partial y} = x+\frac{\partial h(x,y)}{\partial y}[/tex] and so h(x,y) is constant but why does this mean that h only depends on z?

Thanks if you can help.

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