Exam 13: Vector Calculus

Each vector field F graphed below is shown in the xy-plane and looks the same in all other horizontal planes. Is div F at the point P positive, negative or zero? Is curl F at the point P equal to 0? Explain.(a) (b) (c)

Let $\mathbf { F } ( x , y , z ) = z ^ { 2 } \mathbf { i } + x ^ { 2 } \mathbf { j } + y ^ { 2 } \mathbf { k }$ . Find the curl of F.

Evaluate the line integral $\int _ { C } y z d y + x y d z$ , where $C$ is the curve $x = \sqrt { t } , y = t , z = t ^ { 2 } , 0 \leq t \leq 1$ .

Determine whether $\mathbf { F } ( x , y , z ) = \langle x - y , y , z \rangle$ is conservative and if so, find a potential function.

Evaluate the line integral $\int _ { C } y z d x + x z d y + x y d z$ , where $C$ consists of line segments from $( 0,0,0 )$ to $( 2,0,0 )$ , from $( 2,0,0 )$ to $( 1,3 , - 1 )$ , and from $( 1,3 , - 1 )$ to $( 1,3,0 )$ .

Evaluate the line integral $\int _ { C } \left( y + x ^ { 2 } \right) d x + \left( 2 x + \sqrt [ 3 ] { \sin y ^ { 3 } + e ^ { y ^ { 2 } } } \right) d y$ , where C is the upper half of the circle $x ^ { 2 } + y ^ { 2 } = 1$ from $( 1,0 )$ to $( - 1,0 )$ .

Let $\mathbf { F } ( x , y , z ) = \left( z + y ^ { 2 } \right) \mathbf { i } + 2 x y \mathbf { j } + ( x + y ) \mathbf { k }$ . Find the value of the divergence of F at the point $( 1,2,3 )$ .

Evaluate the line integral $\frac { 1 } { 2 } \int _ { C } x d y - y d x$ where $C$ is the curve from $( 0,0 )$ to $( 1,0 )$ , to $( 1,2 )$ to $( 0,2 )$ , then to $( 0,0 )$ .

Evaluate $\iint _ { S } \mathbf { F } \cdot d \mathbf { S }$ , where $\mathbf { F } = ( x , y , z ) = y \mathbf { j } + z \mathbf { k }$ and S is the cube bounded by $x = \pm 1 , \quad y = \pm 1 , \quad z = \pm 1$ .

Sketch the vector field F where F $( x , y ) = x \mathbf { i } - y \mathbf { j }$ .

Let $\mathbf { F } ( x , y , z ) = \left( x ^ { 2 } + y e ^ { x } \right) \mathbf { i } + \left( y ^ { 2 } + z e ^ { x } \right) \mathbf { j } + \left( z ^ { 2 } + x e ^ { y } \right) \mathbf { k }$ and let S be the boundary surface of the solid $E = \left\{ ( x , y , z ) \mid x ^ { 2 } + y ^ { 2 } 1,0 \leq z \leq x + 2 \right\}$ . Evaluate the surface integral $\iint _ { S } \mathbf { F } \cdot d \mathbf { S }$ .

Let $\mathbf { F } ( x , y , z ) = x y i$ and let S be the boundary surface of the solid $E = \{ ( x , y , z ) \mid 0 \leq x \leq 1,0 \leq y \leq 1,0 \leq z \leq 1 \}$ . Evaluate the surface integral $\iint _ { S } \mathbf { F } \cdot d \mathbf { S }$ .

Find the value of the gradient vector field of the function $f ( x , y ) = e ^ { xy ^ { 2 } }$ at the point $( 1,1 )$ .

Evaluate the line integral $\int _ { C } x d y$ along the circle $x = \cos t , y = \sin t , 0 \leq t \leq 2 \pi$ .

Let $\mathbf { F } ( x , y , z ) = x \mathbf { i } + y \mathbf { j } + 2 z \mathbf { k }$ and let S be the boundary surface of the solid $E = \left\{ ( x , y , z ) \mid x ^ { 2 } + y ^ { 2 } \leq z \leq 4 \right\}$ . Evaluate the surface integral $\iint _ { S } \mathbf { F } \cdot d \mathbf { S }$ .

What value of the constant a makes the vector field $\mathbf { F } ( x , y , z ) = 2 x z \mathbf { i } + a y ^ { 2 } z \mathbf { j } + \left( x ^ { 2 } + y ^ { 3 } \right) \mathbf { k }$ conservative?

Evaluate $\iint _ { S } \mathbf { F } \cdot d \mathbf { S }$ , where $\mathbf { F } ( x , y , z ) = x ^ { 2 } \mathbf { i } + x y \mathbf { j } + z \mathbf { k }$ and S is the part of the surface $z = x ^ { 2 } + y ^ { 2 }$ below the plane $z = 1$ , with upward orientation.

Find the mass and center of mass of a thin wire in the shape of a quarter-circle $x ^ { 2 } + y ^ { 2 } = r ^ { 2 }$ , $x \geq 0 , y \geq 0$ , if the density function is $\rho ( x , y ) = x + y$ .

Evaluate the flux of the vector field $\mathbf { F } = 2 \mathbf { i } + \mathbf { j } + 3 \mathbf { k }$ through the plane region with the given orientation as shown below.

Find the flux of $\mathbf { F } ( x , y , z ) = \left( y z ^ { 2 } \right) \mathbf { i } + \left( \mathrm { x } + y ^ { 3 } \right) \mathbf { j } + \left( z ^ { 3 } \right) \mathbf { k }$ across the surface of the solid bounded by the paraboloid $x = 4 - z ^ { 2 } - y ^ { 2 }$ and the $y z -$ plane.