# Find Vector Field Given The Curl

1. Mar 28, 2012

### tazzzdo

1. The problem statement, all variables and given/known data

Find a vector field $\vec{A}$($\vec{r}$) in ℝ3 such that:

$\vec{\nabla} \times \vec{A}$ = y2cos(y)e-y$\hat{i}$ + xsin(x)e-x2$\hat{j}$

3. The attempt at a solution

I broke it down into a series of PDE's that would be the result of $\vec{\nabla} \times \vec{A}$:

∂A3/∂y - ∂A2/∂z = y2cos(y)e-y

∂A3/∂x - ∂A1/∂z = -[-xsin(x)e-x2] (since the j component has a negative sign)

∂A2/∂x - ∂A1/∂y = 0

By a little trial and error I came up with:

A1 = zxsin(x)e-x2
A2 = -zy2cos(y)e-y
A3 = z

This can be verified that:

$\vec{\nabla} \times \vec{A}$ = y2cos(y)e-y$\hat{i}$ + xsin(x)e-x2$\hat{j}$

Is this a valid answer then?

2. Mar 28, 2012

### Dick

Sure it is. Why would you think it isn't?

3. Mar 28, 2012

### tazzzdo

Lol my Vector Calc professor tends to be picky. I didn't know if there was a more "rigorous" way to do it. But if this works, then I'm fine.

4. Mar 28, 2012

### Dick

Trial and error is a perfectly fine way to solve a problem like this.