Exam 14: Vector Analysis

Find the flux of $\vec { F }$ through the surface $S$ in the positive z direction. Let $S = \{ ( u + v , 3 u + 2 v , 6 v ) : 0 \leq u \leq 1 \text { and } 1 \leq v \leq 3 \}$ and $\vec { F } ( x , y , z ) = \langle z , x , y \rangle$

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

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

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

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

Find $\operatorname { Div } \vec { F }$ for $\vec { F } ( x , y , z ) = \langle x y z , x y z , x y z \rangle$

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

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

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

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

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

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

Find $\int _ { C } \vec { F } \cdot d \vec { r }$ of the function over the given curve: $\vec { F } ( x , y ) = \langle 1 , x , 1 \rangle$ and the curve C is given by $\vec { r } ( t ) = \langle \cos ( t ) , \sin ( t ) , t \rangle , 0 \leq t \leq \frac { \pi } { 2 }$

Compute the integral using Green's theorem. $\int _ { C } \vec { F } \cdot d \vec { r }$ , where C is the square with vertices (1, 1), (0, 1), (0, 0), and (1, 0) traversed counterclockwise and $\vec { F } ( x , y ) = \left\langle x ^ { 2 } y , x ^ { 2 } y ^ { 2 } \right\rangle$

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

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

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

Find the area of 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 $0 \leq y \leq 4$

Compute the integral using Green's theorem. $\int _ { C } \vec { F } \cdot d \vec { r }$ , where C is the unit circle traversed counterclockwise and $\vec { F } ( x , y ) = \left\langle y ^ { 3 } , - x ^ { 3 } \right\rangle$

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 $\vec { F } ( x , y , z ) = \langle x z , y x , x y z \rangle$