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I'm studying out of Stewart's for my Calc IV class, and hit a stumbling block in his section on the fundamental theorem for line integrals. He shows a process of finding a potential function [tex]f[/tex] such that [tex]\vec{F} = \nabla f [/tex], where [tex]\vec{F} [/tex] is a vector field defined as

[tex]\vec{F} = (3 + 2xy) \mathbf{\hat{i}} + (x^2 - 3y^2) \mathbf{\hat{j}} [/tex]

He then goes on to equate the components of the gradient vector of the function we want to find with the components of; with the given vector field.

[tex]\frac{\partial f(x,y)}{\partial x} = 3 + 2xy [/tex] (eq. 7)

[tex]\frac{\partial f(x,y)}{\partial y} = x^2 - 3y^2 [/tex] (eq.8)

no problems so far.

but now he integrates equation 7 with respect to x and obtains:

[tex]f(x,y) = 3x + x^2y + g(y) [/tex]

He doesn't really explain where g(y) comes from, or why it is needed. I know a constant of integration is needed, but why must it be a function of y? Thanks for the help.

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# Partial integration of gradient vector to find potential field

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