Find Vector Field Given The Curl

[tazzzdo]

Homework Statement

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$\hat{j}$

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?

 

[Dick]

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

 

[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.

 

[Dick]

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.

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