Exam 14: Vector Analysis

Compute the integral using Green's theorem. $\int _ { C } \vec { F } \cdot d \vec { r }$ , where C is the triangle with vertices (1, 1), (0, 0), and (1, 0) traversed counterclockwise and $\vec { F } ( x , y ) = \left\langle x ^ { 2 } y , x ^ { 3 } y ^ { 3 } \right\rangle$

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

Integrate the function $f$ over 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 $2 \leq x \leq 4$ and $f ( x , y ) = x ^ { 2 }$

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

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

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

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

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

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

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

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 $\vec { F } ( x , y , z ) = \langle z , y , 2 x \rangle$

Is the given vector field conservative? $\vec { F } ( x , y ) = \left\langle x ^ { y - 1 } , x ^ { y } \ln ( x ) \right\rangle$

Find the area of 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 $1 \leq y \leq 6$

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

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