Exam 14: Vector Analysis

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

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

Integrate the function $f$ over the surface $S$ Let $S = \{ ( 2 u + v , u - v , u + 3 v ) : 0 \leq u \leq 2 \text { and } 0 \leq v \leq 1 \}$ and $f ( x , y ) = e ^ { x }$

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

Find the divergence of $\vec { F } ( x , y , z ) = \left\langle x , y ^ { 2 } , z ^ { 3 } \right\rangle$

Integrate the function $f$ over the surface $S$ Let $S = \{ ( u + v , 3 u + 2 v , 6 v ) : 0 \leq u \leq 1 \text { and } 1 \leq v \leq 3 \}$ and $f ( x , y ) = \sin ( x )$

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 $y = x ^ { 3 } , 0 \leq x \leq 2$

Is the given vector field conservative? $\vec { F } ( x , y ) = \left\langle 3 x y ^ { 3 } , 3 x ^ { 2 } y ^ { 3 } \right\rangle$

Find a potential for the given vector field: $\vec { F } ( x , y ) = \left\langle 3 x ^ { 2 } \cos ( x y ) - x ^ { 3 } y \sin ( x y ) , - x ^ { 4 } \sin ( x y ) \right\rangle$

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

Find the area of the surface $S$ Let $S = \{ ( v , u , u ) : 0 \leq u \leq 1 \text { and } 0 \leq v \leq 1 \}$

Find the area of the surface $S$ Let $S = \{ ( u + v , 3 u + 2 v , 6 v ) : 0 \leq u \leq 1 \text { and } 1 \leq v \leq 3 \}$

Find the divergence of $\vec { F } ( x , y ) = \left\langle \sin ( x y ) , \cos \left( y ^ { 2 } \right) \right\rangle$

Is the given vector field conservative? $\vec { F } ( x , y ) = \left\langle 2 x y ^ { 3 } , 3 x ^ { 2 } y ^ { 2 } \right\rangle$

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 square with vertices (0, 0), (2, 0), (2, 2), and (0, 2) with the normal vector in the positive z direction where $\vec { F } ( x , y , z ) = \langle x z , y x , x y z \rangle$

Evaluate the line integral of the function over the given curve: $f ( x , y ) = x y$ 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 the boundary curve of the surface $S = \{ ( u v , u , u - v ) : 0 \leq u \leq 1 \text { and } 0 \leq v \leq 1 \}$ with the normal vector in the positive z direction where $\vec { F } ( x , y , z ) = \langle x , x , y \rangle$

Evaluate the line integral of the function over the given curve: $f ( x , y ) = x e ^ { y }$ and the curve C is given by $\vec { r } ( t ) = \langle \cos ( t ) , \sin ( t ) \rangle , 0 \leq t \leq \frac { \pi } { 2 }$

Evaluate the line integral of the function over the given curve: $f ( x , y ) = x$ and the curve C is given by $\vec { r } ( t ) = \langle \cos ( t ) , \sin ( t ) \rangle , 0 \leq t \leq \frac { \pi } { 2 }$

Evaluate the line integral of the function over the given curve: $f ( x , y , z ) = y e ^ { x }$ and the curve C is given by $\vec { r } ( t ) = \langle \cos ( t ) , \sin ( t ) , t \rangle , 0 \leq t \leq \frac { \pi } { 2 }$