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Homework Statement
Use Stokes' Theorem to evaluate [itex]\int\int curl \vec{F}\bullet d\vec{S} [/itex] where [itex]\vec{F}(x,y,z) = <e^{z^{2}},4z-y,8xsin(y)> [/itex] and S is the portion of the paraboloid
[itex]z = 4-x^{2}-y^{2}[/itex] above the xy plane.
Homework Equations
Stokes Thm:[itex]\int\int curl \vec{F}\bullet d\vec{S} = \int \vec{F}\bullet d\vec{r}[/itex]
[itex]\vec{F}(x,y,z) = <e^{z^{2}},4z-y,8xsin(y)> [/itex]
S: [itex]z = 4-x^{2}-y^{2}[/itex] above the z = 0.
The Attempt at a Solution
C: [itex]\vec{r}(t) = <2cos(t), 2sin(t), 0>[/itex] where [itex] 0\leq t\leq2\pi [/itex]
[itex]\vec{r}'(t) = <-2sin(t), 2cos(t), 0> [/itex]
[itex]\vec{F}(\vec{r}(t)) = <e^{0}, (4(0) - 2sin(t), 8(2(cos(t))sin(2cos(t))>[/itex]
[itex]\vec{F}(\vec{r}(t)) = <1, -2sin(t), 16cos(t)sin(2cos(t))>[/itex]
[itex]\int <1, -2sin(t), 16cos(t)sin(2cos(t))> \bullet <-2sin(t), 2cos(t), 0> dt [/itex] from 0 to 2pi
[itex]=\int -2sin(t)-2sin(t)2cos(t) dt [/itex] from 0 to 2pi
[itex]=0 [/itex]
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