Exam 16: Vector Calculus

The area of the surface $\mathbf { r } ( u , v ) = ( 2 + \cos v ) \cos u \mathbf { i } + ( 2 + \cos v ) \sin u \mathbf { j } + \sin v \mathbf { k }$ , where $0 \leq u \leq 2 \pi$ and $0 \leq v \leq 2 \pi$ is

If I = $\int$ C $\left( e ^ { - y } - 2 x \right) d x - \left( x e ^ { - y } + \sin y \right) d y$ is independent of the path where C is a curve from (0, ?) to (?, 0), then I is

The rectangular equation for the parametric surface and $x ( u , v ) = 4 \cos v \cos u$ is

The line integral $\int$ C $3 x d x + 2 x y d y + d z$ where C is the curve is

Using Green's Theorem, the line integral $\int$ C $y ^ { 2 } d x + x ^ { 2 } d y$ where C is the closed curve consisting of the arc of $y = x ^ { 2 }$ from (0, 0) to (1, 1), the line segment from (1, 1) to (1, 0), and the line segment from (1, 0) to (0, 0), is

Let $f ( x , y , z ) = \sqrt { x ^ { 2 } + 3 y z - z ^ { 2 } }$ . Its gradient vector field is

If I = $\int$ C $3 \left( 2 x ^ { 2 } + 6 x y \right) d x + 3 \left( 3 x ^ { 2 } + 8 \right) d y$ is independent of the path where C is a curve from (1, 0) to (0, 1) then I is

The surface integral $\int \int$ S $y d S$ where S is $z = 4 - y ^ { 2 }$ in the first octant bounded by $x = 3$ and the coordinate planes, is

The line integral $\int$ C $x y ^ { 2 } d s$ where C is the curve $x = \cos t , y = \sin t , 0 \leq t \leq \frac { \pi } { 2 }$ , is

Let $\mathbf { F } ( x , y , z ) = x \mathbf { i } + y \mathbf { j } + z \mathbf { k }$ and S is the boundary of the region enclosed on the side by $x ^ { 2 } + y ^ { 2 } = 9$ , below by z = 0, and above z = 4. Using the Divergence Theorem, $\iint S$$\mathbf { F } \bullet \mathbf { n } d S$ is

The rectangular equation for the parametric surface $x ( u , v ) = u$ $y ( u , v ) = v$ and $z ( u , v ) = \frac { v } { 3 }$ is

The outer unit normal vector to $x + y + z = 1$ is

Let $\mathbf { F } ( x , y ) = 2 x y \mathbf { i } + 3 x \mathbf { j } , \mathbf { r } ( t ) = 3 t ^ { 2 } \mathbf { i } - t \mathbf { j } , 0 \leq t \leq 1$ . Then $\int$ C $\mathbf { F } \bullet d \mathbf { r }$ is

The work done by the force $\mathbf { F } ( x , y ) = - x ^ { 2 } y \mathbf { i } + 2 y \mathbf { j }$ moving along $\mathbf { r } ( t ) = \cos t \mathbf { i } + \sin t \mathbf { j }$ with $0 \leq t \leq \frac { \pi } { 2 }$ is

Let $\mathbf { F } ( x , y , z ) = 5 z \mathbf { k }$ and S is $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 16$ Using the Divergence Theorem, $\iint S$ $\mathbf { F } \bullet \mathbf { n } d S$ is

Using Green's Theorem, the line integral $\int$ C $\left( x ^ { 2 } - y ^ { 2 } \right) d x + 2 x y d y$ where C is the circle $x ^ { 2 } + y ^ { 2 } = 1 ,$ is

Let $\mathbf { F } ( x , y , z ) = e ^ { x } \mathbf { i } + 3 x ^ { 2 } y \mathbf { j } + \left( 2 y ^ { 2 } z + x \right) \mathbf { k }$ . Then curl F is

Let $\mathbf { F } ( x , y , z ) = x \mathbf { i } + y \mathbf { j } + z \mathbf { k }$ and S is $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 4$ . Using the Divergence Theorem, $\iint S$$\mathbf { F } \bullet \mathbf { n } d S$ is

Let $\mathbf { F } ( x , y , z ) = \cos y \mathbf { i } + \cos z \mathbf { j } + \cos x \mathbf { k }$ . Then curl F is

Let $\mathbf { F } ( x , y , z ) = x ^ { 2 } \mathbf { i } + y ^ { 2 } \mathbf { j } + z ^ { 2 } \mathbf { k }$ and Sis the region bounded by and $x = 0 , x = 3$ . Using the Divergence Theorem, $\iint S$$y = 0 , y = 3 \text {, }$ is