$\text { Use Stokes's Theorem to evaluate } \int _ { C } \mathbf { F } \cdot d r \text { where } \mathbf { F } ( x , y , z ) = x ^ { 2 } \mathbf { i } + z ^ { 2 } \mathbf { j } - x y z \mathbf { k }$ and $S$ is $z = \sqrt { 25 - x ^ { 2 } - y ^ { 2 } }$.Use a computer algebra system to verify your result.
A) 10
B) 9
C) 11
D) 7
E) 0

Question

Quizplus · Accepted Answer

Stokes's Theorem relates a surface integral over a surface $S$ to a line integral over the boundary curve $C$ of $S$. The theorem states that $\int _ { C } \mathbf { F } \cdot d \mathbf { r } = \int _ { S } \nabla \times \mathbf { F } \cdot d \mathbf { S }$, where $\nabla \times \mathbf { F }$ is the curl of $\mathbf { F }$. For the given $\mathbf { F } = x ^ { 2 } \mathbf { i } + z ^ { 2 } \mathbf { j } - x y z \mathbf { k }$, the curl $\nabla \times \mathbf { F }$ is $\mathbf { 0 }$, because the partial derivatives of the components of $\mathbf { F }$ with respect to $x$, $y$, and $z$ do not vary in a way that would produce a non-zero curl. Therefore, the integral over any surface bounded by $C$ is $0$, indicating that there is no circulation of $\mathbf { F }$ around $C$.

And SSS Is z=25−x2−y2z = \sqrt { 25 - x ^ { 2 } - y ^ { 2 } }z=25−x2−y2​

And $S$ Is $z = \sqrt { 25 - x ^ { 2 } - y ^ { 2 } }$