Use Stokes's Theorem to evaluate $\mathbf { F } ( x , y , z ) = x y z \mathbf { i } + y \mathbf { j } + z \mathbf { k } , x ^ { 2 } + y ^ { 2 } \leq 49$ and $S$ is the first-octant portion of $z = x ^ { 2 }$ over $x ^ { 2 } + y ^ { 2 } = 49$
Use a computer algebra system to verify your result.
A) $\frac { 16,807 } { 15 }$
B) $- \frac { 16,807 } { 15 }$
C) $- \frac { 33,614 } { 15 }$
D) 0
E) $\frac { 33,614 } { 15 }$

Question

Use Stokes's Theorem to evaluate $\mathbf { F } ( x , y , z ) = x y z \mathbf { i } + y \mathbf { j } + z \mathbf { k } , x ^ { 2 } + y ^ { 2 } \leq 49$ and $S$ is the first-octant portion of $z = x ^ { 2 }$ over $x ^ { 2 } + y ^ { 2 } = 49$
Use a computer algebra system to verify your result.
A) $\frac { 16,807 } { 15 }$
B) $- \frac { 16,807 } { 15 }$ 
C) $- \frac { 33,614 } { 15 }$
D) 0
E) $\frac { 33,614 } { 15 }$

Quizplus · Accepted Answer

Stokes's Theorem gives$$\begin{split} \oint\limits_C \mathbf{F} \cdot d\mathbf{r} & = \iint\limits_S (
abla 	imes \mathbf{F}) \cdot \mathbf{n} \ d\sigma \\ & = \iint\limits_S \left[ \left(\frac{\partial P}{\partial y} - \frac{\partial N}{\partial z}ight) \mathbf{i} + \left(\frac{\partial M}{\partial z} - \frac{\partial P}{\partial x}ight) \mathbf{j} + \left(\frac{\partial N}{\partial x} - \frac{\partial M}{\partial y}ight) \mathbf{k} ight] \cdot \mathbf{n} \ d\sigma \\ & = \iint\limits_S (y-y) \mathbf{i} + (z-x) \mathbf{j} + (x-z) \mathbf{k} \cdot \mathbf{n} \ d\sigma \\ & = \iint\limits_S (z-x) \mathbf{j} + (x-z) \mathbf{k} \cdot \mathbf{n} \ d\sigma \\ & = \iint\limits_S (z-x) \left(\frac{\partial}{\partial x} [x^2] + \frac{\partial}{\partial y} [0] ight) + (x-z) \left(\frac{\partial}{\partial x} [0] + \frac{\partial}{\partial y} [x^2] ight) \ d\sigma \\ & = \iint\limits_S (z-x)(2x) + (x-z)(2y) \ d\sigma \\ & = \iint\limits_S 2(x^2 -xz + xy - yz) \ d\sigma \\ & = \iint\limits_S 2(x^2 -xz + xy - yz) \ d\sigma \\ & = \int_0^{2\pi} \int_0^7 2(r^2 \cos^2 	heta - r^2 \cos 	heta \sin 	heta + r^2 \cos 	heta \sin 	heta - r^2 \sin^2 	heta) \ r \ dr \ d	heta \\ & = \int_0^{2\pi} \int_0^7 2(r^3 \cos^2 	heta - r^3 \cos 	heta \sin 	heta + r^3 \cos 	heta \sin 	heta - r^3 \sin^2 	heta) \ dr \ d	heta \\ & = \int_0^{2\pi} \int_0^7 2(r^3 \cos^2 	heta - r^3 \sin^2 	heta) \ dr \ d	heta \\ & = \int_0^{2\pi} \int_0^7 2r^3 (\cos^2 	heta - \sin^2 	heta) \ dr \ d	heta \\ & = \int_0^{2\pi} \int_0^7 2r^3 \cos 2	heta \ dr \ d	heta \\ & = \int_0^{2\pi} \left[ \frac{r^4}{2} \cos 2	heta ight]_0^7 \ d	heta \\ & = \int_0^{2\pi} \frac{2401}{2} \cos 2	heta \ d	heta \\ & = \left[ \frac{2401}{4} \sin 2	heta ight]_0^{2\pi} \\ & = \frac{2401}{4} (0-0) = 0 \end{split}$$

Use Stokes's Theorem to Evaluate F(x,y,z)=xyzi+yj+zk,x2+y2≤49\mathbf { F } ( x , y , z ) = x y z \mathbf { i } + y \mathbf { j } + z \mathbf { k } , x ^ { 2 } + y ^ { 2 } \leq 49F(x,y,z)=xyzi+yj+zk,x2+y2≤49

Use Stokes's Theorem to Evaluate $\mathbf { F } ( x , y , z ) = x y z \mathbf { i } + y \mathbf { j } + z \mathbf { k } , x ^ { 2 } + y ^ { 2 } \leq 49$