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Find Vector Field Given The Curl

  1. Mar 28, 2012 #1
    1. The problem statement, all variables and given/known data

    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]

    3. 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?
     
  2. jcsd
  3. Mar 28, 2012 #2

    Dick

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    Sure it is. Why would you think it isn't?
     
  4. Mar 28, 2012 #3
    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.
     
  5. Mar 28, 2012 #4

    Dick

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