- #1

tazzzdo

- 47

- 0

## Homework Statement

Find a vector field [itex]\vec{A}[/itex]([itex]\vec{r}[/itex]) in ℝ

^{3}such that:

[itex]\vec{\nabla} \times \vec{A}[/itex] = y

^{2}cos(y)e

^{-y}[itex]\hat{i}[/itex] + xsin(x)e

^{-x2}[itex]\hat{j}[/itex]

## The Attempt at a Solution

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

∂A

_{3}/∂y - ∂A

_{2}/∂z = y

^{2}cos(y)e

^{-y}

∂A

_{3}/∂x - ∂A

_{1}/∂z = -[-xsin(x)e

^{-x2}] (since the j component has a negative sign)

∂A

_{2}/∂x - ∂A

_{1}/∂y = 0

By a little trial and error I came up with:

A

_{1}= zxsin(x)e

^{-x2}

A

_{2}= -zy

^{2}cos(y)e

^{-y}

A

_{3}= z

This can be verified that:

[itex]\vec{\nabla} \times \vec{A}[/itex] = y

^{2}cos(y)e

^{-y}[itex]\hat{i}[/itex] + xsin(x)e

^{-x2}[itex]\hat{j}[/itex]

Is this a valid answer then?