WebWork 20.3 #7

Problem Statement: Three small squares, $S_1$, $S_2$, and $S_3$, each with side length 0.1 and centered at the point $(5,4,3)$, lie parallel to the $xy$-, $yz$-, and $xz$-planes, respectively. The squares are oriented counterclockwise when viewed from the positive $z$-, $x$-, and $y$-axes, respectively. A vector field $\vec{G}$ has circulation around $S_1$ of $2$, around $S_2$ of $-0.05$, and around $S_3$ of $-3$. Estimate $\text{curl}\vec{G}$ at the point $(5,4,3)$.

Hints: The main hint here is that we know


Thus, if we consider a normal vector to the surface $S_1$ (for instance), knowing that the surface is oriented counterclockwise when viewed from the positive $z$-axis leads us to conclude that our normal vector for $S_1$ is simply $\vec{k}$. Since we know that the circulation is $2$ around $S_1$, we now know that


By definition of circulation density and the known value for circulation on $S_1$, we can then write

\[\text{curl}\vec{G}\cdot\vec{k}\approx \frac{\int_{S_1}\vec{G}\cdot d\vec{r}}{\text{Area}(S_1)}=\frac{2}{0.1^2}.\]

The main thing to realize here is that $\text{curl}\vec{G}\cdot\vec{k}$ is simply the $\vec{k}$ component of $\text{curl}\vec{G}$. So, if one performs the same calculation for each component, the calculation of finding an approximation to $\text{curl}\vec{G}$ is done.


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