The Fundamental Theorem for Line Integrals

[Chas3down]

Homework Statement

Determine whether or not f(x,y) is a conservative vector field.
f(x,y) = <-3e^(-3x)sin(-3y),-3e^(-3x)cos(-3y) >

If F is a conservative fector field find F = gradient of f

Homework Equations

N/A

The Attempt at a Solution

Fx = -3e^(-3x)(-3)cos(-3y)
Fy = -3e^(-3x)(-3)cos(-3y)

f is a conservative fector field
(This part is all correct.)

F = ? + K

It won't accept a vector, I know how to normally find a gradient vector, but that returns a vector, I need a non-vector answer..

 

[ShayanJ]

$\vec \nabla f=<\partial_x f,\partial_y f>=\vec F=<F_x,F_y> \Rightarrow \left\{ \begin{array}{c} f=\int F_x dx \\ f=\int F_y dy \end{array} \right.$

 

[STEMucator]

Since $\vec F(x,y)$ is conservative, you know $\vec F(x,y) = \vec{\nabla f(x,y)}$ for some potential function $f$.

This amounts to saying:

$\vec F(x,y) = \vec{\nabla f(x,y)}$
$P \hat i + Q \hat j = f_x \hat i + f_y \hat j$

Equating the vector components: $P = f_x$ and $Q = f_y$.

So really you want to solve those two equations for some $f(x,y)$. The usual method would be to integrate one of those two equations to find some $f(x,y)$ that has a constant of integration which varies. For example, taking the first:

$$f(x,y) = \int P dx = P' + g(y)$$

Afterwards, you should look at the equation you didn't use and observe the outcome.