(adsbygoogle = window.adsbygoogle || []).push({}); "partial integration" of gradient vector to find potential field

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.

**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!

# Partial integration of gradient vector to find potential field

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