Proving F = grad(f) with 3 Variables

[jwxie]

Homework Statement

For two variables, F = grad(f) if only if partial P / partial y is equal to partial Q / partial x
where, P and Q represent the x, y function, P(x,y) and Q(x,y)

For example:
F(x,y) = P(x,y) + Q(x,y)

Now the question is, for three variables, I have P + Q + R.
Is proving partial P / partial y = partial Q / partial x enough to say that F = grad(f)? If not, what is the approach to prove it for a 3 variables?

Thank you

 

[ystael]

The condition for a vector field in three-space to be the gradient of some scalar field is a generalization of the condition in the plane: the vector field must have zero curl, $$\nabla\times\mathbf{F} = 0$$. This works out to three equations, not one, between the partial derivatives of the components of $$\mathbf{F}$$: $$\frac{\partial F_x}{\partial y} - \frac{\partial F_y}{\partial x} = \frac{\partial F_x}{\partial z} - \frac{\partial F_z}{\partial x} = \frac{\partial F_z}{\partial y} - \frac{\partial F_y}{\partial z} = 0$$.

The proof of this fact uses the same argument as in the plane case (a gradient vector field is one whose integral around a closed loop is always zero, so you can construct a potential function $$f$$ with $$\mathbf{F} = \nabla f$$ by integration) together with Stokes's theorem.

 

[jwxie]

Thank you very much!