- #1
JJBladester
Gold Member
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Homework Statement
Determine the curl of the vector function below.
[tex]\boldsymbol{F}\left ( x,y,z \right )=3x^2\boldsymbol{i}+7e^xy\boldsymbol{j}[/tex]
Homework Equations
curl[itex]\mathbf{F}=\mathbf{\nabla}\times \mathbf{F}[/itex]
[tex]=\begin{vmatrix}
\mathbf{i}& \mathbf{j}& \mathbf{k}\\
\frac{\partial}{\partial x}& \frac{\partial}{\partial y}& \frac{\partial}{\partial z}\\
P(x,y,z)& Q(x,y,z)& R(x,y,z)
\end{vmatrix}[/tex]
The Attempt at a Solution
This problem is solved by my FE review as below. I understand how to solve for the determinant of a 3x3 matrix by rewriting the first two columns to the right of the matrix and obtaining three "+" terms and three "-" terms. I think my partials for x and y are off, perhaps.
[tex]\mathbf{i}\left ( \frac{\partial }{\partial y}0-\frac{\partial }{\partial z}7e^xy \right )-\mathbf{j}\left ( \frac{\partial }{\partial x}0-\frac{\partial }{\partial z}3x^2 \right )+\mathbf{k}\left ( \frac{\partial }{\partial x}7e^xy-\frac{\partial }{\partial y}3x^2 \right )[/tex]
[tex]=\mathbf{i}(0-0)-\mathbf{j}(0-0)-\mathbf{k}\left ( 7e^xy-0 \right )=7e^xy\mathbf{k}[/tex]
The expressions I calculated for [itex]\mathbf{i} and \mathbf{j}[/itex] match what the book has. However, my expression for [itex]\mathbf{k}[/itex] seems to be incorrect. Here is what I calculated for the values of the matrix:
[tex]\frac{\partial }{\partial x}=6x[/tex]
[tex]\frac{\partial }{\partial y}=7e^x[/tex]
[tex]\frac{\partial }{\partial z}=0[/tex]
[tex]P(x,y,z)=3x^2[/tex]
[tex]Q(x,y,z)=7e^xy[/tex]
[tex]R(x,y,z)=0[/tex]
So my expression for [itex]\mathbf{k}[/itex] was:
[tex]\left [(6x)7e^xy-\left ( 7e^x \right )3x^2 \right ]\mathbf{k}[/tex]
I think I went wrong with my calculation of [itex] \frac{\partial }{\partial x}[/itex].