Find Vector Field Given The Curl

  • Thread starter tazzzdo
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  • #1
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Homework Statement



Find a vector field [itex]\vec{A}[/itex]([itex]\vec{r}[/itex]) in ℝ3 such that:

[itex]\vec{\nabla} \times \vec{A}[/itex] = y2cos(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]:

∂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:

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

Is this a valid answer then?
 

Answers and Replies

  • #2
Dick
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Sure it is. Why would you think it isn't?
 
  • #3
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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
Dick
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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.
 

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