Exam 14: Vector Analysis

Integrate the function $f$ over the surface $S$ Let $S$ be the portion of the plane $z = 2 x + y + 3$ above the rectangle $0 \leq x \leq 2$ and $1 \leq x \leq 6$ and $f ( x , y ) = x$

Use Stokes' theorem to evaluate $\int _ { C } \vec { F } \cdot d \vec { r }$ , where C is in the plane $z = 6 - x - y$ lying above the square with vertices (0, 0), (1, 0), (1, 1), and (0, 1) with the normal vector in the positive z direction where $\vec { F } ( x , y , z ) = \langle z , y , 2 x \rangle$

Compute $\iint _ { S } \vec { F } \cdot \vec { n } d S$ where S is the unit sphere, $\vec { n }$ is the unit outward normal, and $\vec { F } ( x , y , z ) = \left\langle x ^ { 2 } , y , 4 \right\rangle$

Find a potential for the given vector field: $\vec { F } ( x , y ) = \left\langle 2 x y ^ { - 1 } , - x ^ { 2 } y ^ { - 2 } \right\rangle$

Find a potential for the given vector field: $\vec { F } ( x , y , z ) = \left\langle y z e ^ { x y z } , x z e ^ { x y z } , x y e ^ { x y z } \right\rangle$

Find the curl of $\vec { F } ( x , y , z ) = \left\langle e ^ { y z } , e ^ { x z } , e ^ { x y } \right\rangle$

Use Stokes' theorem to evaluate $\int _ { C } \vec { F } \cdot d \vec { r }$ , where C is in the plane $z = 6 - x - y$ lying above the square with vertices (0, 0), (1, 0), (1, 1), and (0, 1) with the normal vector in the positive z direction where $\vec { F } ( x , y , z ) = \left\langle z , y ^ { 2 } , 3 x \right\rangle$

Find the divergence of $\vec { F } ( x , y ) = \langle x , x y \rangle$

Integrate the function $f$ over the surface $S$ Let $S = \{ ( v , u , u ) : 0 \leq u \leq 1 \text { and } 0 \leq v \leq 1 \}$ and $f ( x , y ) = y$

Find the divergence of $\vec { F } ( x , y , z ) = \langle x y , y z , x z \rangle$

Find the area of the surface $S$ Let $S$ be the portion of the plane $z = x + 9 y - 2$ above the rectangle $1 \leq x \leq 3$ and $2 \leq y \leq 4$

Evaluate the line integral of the function over the given curve: $f ( x , y ) = y$ and the curve C is given by $y = x ^ { 3 } , 0 \leq x \leq 2$

Compute $\iint _ { S } \vec { F } \cdot \vec { n } d S$ where S is the unit sphere, $\vec { n }$ is the unit outward normal, and $\vec { F } ( x , y , z ) = \left\langle x ^ { 3 } , y ^ { 3 } , z ^ { 3 } \right\rangle$

Find $\int _ { C } \vec { F } \cdot d \vec { r }$ of the function over the given curve: $\vec { F } ( x , y ) = \langle x , y \rangle$ and the curve C is given by $\vec { r } ( t ) = \langle \cos ( t ) , \sin ( t ) \rangle , 0 \leq t \leq \frac { \pi } { 2 }$

Use Stokes' theorem to evaluate $\int _ { C } \vec { F } \cdot d \vec { r }$ , where C is in the plane $z = 7 - 2 x - y$ lying above the unit disk with the normal vector in the positive z direction where $\vec { F } ( x , y , z ) = \langle z , y , 2 x \rangle$

Integrate the function $f$ over the surface $S$ Let $S$ be the portion of the plane $z = 3 x + 4 y + 7$ above the rectangle $0 \leq x \leq 3$ and $0 \leq x \leq 4$ and $f ( x , y ) = x y$

Find $\operatorname { Div } \vec { F }$ for $\vec { F } ( x , y , z ) = \left\langle x ^ { 2 } e ^ { y } , y ^ { 2 } e ^ { x } , y \right\rangle$

Find $\int _ { C } \vec { F } \cdot d \vec { r }$ of the function over the given curve: $\vec { F } ( x , y ) = \langle y , y \rangle$ and the curve C is given by $\vec { r } ( t ) = \langle \cos ( t ) , \sin ( t ) \rangle , 0 \leq t \leq \frac { \pi } { 2 }$

Find $\int _ { C } \vec { F } \cdot d \vec { r }$ of the function over the given curve: $\vec { F } ( x , y ) = \langle x , x \rangle$ and the curve C is given by $\vec { r } ( t ) = \langle \cos ( t ) , \sin ( t ) \rangle , 0 \leq t \leq \frac { \pi } { 2 }$

Find the flux of $\vec { F }$ through the surface $S$ in the positive z direction. Let $S = \{ ( 2 u + v , u - v , u + 3 v ) : 0 \leq u \leq 2 \text { and } 0 \leq v \leq 1 \}$ and $\vec { F } ( x , y , z ) = \langle x , y , z - y \rangle$