Exam 17: Integrals and Vector Fields

Use Stokes' Theorem to calculate the circulation of the field F around the curve C in the indicated direction. - $\mathbf { F } = 3 x \mathbf { i } + 2 x \mathbf { j } + 7 \mathrm { zk } ; \mathrm { C }$ : the cap cut from the upper hemisphere $\mathrm { x } ^ { 2 } + \mathrm { y } ^ { 2 } + \mathrm { z } ^ { 2 } = 16 ( \mathrm { z } \geq 0 )$ by the cylinder $\mathrm { x } ^ { 2 } + \mathrm { y } ^ { 2 } = 4$

Evaluate the work done between point 1 and point 2 for the conservative field F. - $\mathbf { F } = 16 x e ^ { 8 x ^ { 2 } - 9 y ^ { 2 } - 2 z ^ { 2 } } \mathbf { i } - 18 y e ^ { 8 x ^ { 2 } - 9 y ^ { 2 } - 2 z ^ { 2 } } \mathbf { j } - 4 z e ^ { 8 x ^ { 2 } - 9 y ^ { 2 } - 2 z ^ { 2 } } \mathbf { k } ; P _ { 1 } ( 0,0,0 ) , P _ { 2 } ( 1,1,1 )$

Evaluate. The differential is exact. - $\int _ { ( 0,0,0 ) } ^ { ( 1,1,1 ) } \left( 8 x - 10 x ^ { 9 } y ^ { 5 } \right) d x - 5 x ^ { 10 } y ^ { 4 } d y - 45 z ^ { 4 } d z$

Solve the problem. -The shape and density of a thin shell are indicated below. Find the moment of inertia about the $z$ -axis. Shell: "nose" of the paraboloid $x ^ { 2 } + y ^ { 2 } = 2 z$ cut by the plane $z = 2$ Density: $\delta = \frac { 1 } { \sqrt { x ^ { 2 } + y ^ { 2 } + 1 } }$

Find the surface area of the surface S. - $S$ is the paraboloid $x ^ { 2 } + y ^ { 2 } - z = 0$ between the planes $z = 0$ and $z = 6$ .

Find the equation for the plane tangent to the parametrized surface S at the point P. - $S \text { is the cylinder } r ( \theta , z ) = 6 \cos ^ { 2 } \theta \mathbf { i } + 3 \sin 2 \theta \mathbf { j } + \mathrm { zk } ; \mathrm { P } \text { is the point corresponding to } ( \theta , z ) = \left( \frac { \pi } { 4 } , - 1 \right)$

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Sketch the vector field in the plane along with its horizontal and vertical components at a representative assortment of points on the circle $x ^ { 2 } + y ^ { 2 } = 4$ . -F = xi - yj

Evaluate the surface integral of G over the surface S. - $S$ is the hemisphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 3 , z \geq 0 ; G ( x , y , z ) = z ^ { 2 }$

Find the equation for the plane tangent to the parametrized surface S at the point P. - $\text { S is the cylinder } r ( \theta , z ) = 8 \cos \theta \mathbf { i } + 8 \sin \theta \mathbf { j } + \mathrm { zk } ; \mathrm { P } \text { is the point corresponding to } ( \theta , \mathrm { z } ) = \left( \frac { \pi } { 2 } , 3 \right) \text {. }$

Use Stokes' Theorem to calculate the circulation of the field F around the curve C in the indicated direction. - $\mathbf { F } = x ^ { 2 } \mathbf { i } + x y \mathbf { j } + y \mathbf { k } ; \mathrm { C }$ : the counterclockwise path around the perimeter of the rectangle in the $x - y$ plane formed from the $x$ -axis, $y$ -axis, $x = 2$ and $y = 3$

Find the gradient field of the function. - $f ( x , y , z ) = \ln \left( x ^ { 4 } + y ^ { 5 } + z ^ { 6 } \right)$

Using Green's Theorem, compute the counterclockwise circulation of F around the closed curve C. -F = (x - y)i + (x + y)j; C is the triangle with vertices at (0, 0), (4, 0), and (0, 9)

Use Stokes' Theorem to calculate the circulation of the field F around the curve C in the indicated direction. - $\mathbf { F } = - 3 \mathrm { y } \mathbf { i } + 5 \mathrm { x } \mathbf { j } + 6 \mathrm { z } ^ { 3 } \mathbf { k } ; \mathrm { C } : \text { the portion of the plane } 8 x + 5 y + 8 z = 10 \text { in the first quadrant }$

Calculate the flow in the field F along the path C. - $\mathbf { F } = \mathrm { e } ^ { 2 } \mathrm { i } + \frac { 1 } { \mathrm { y } } \mathrm { j } + 2 \mathbf { k } , \mathrm { C }$ is the curve $\mathbf { r } ( \mathrm { t } ) = 4 \mathrm { t } ^ { 2 } \mathrm { i } + 5 \mathrm { tj } + ( - 8 - 2 \mathrm { t } ) \mathbf { k } , 1 \leq \mathrm { t } \leq 3$

Test the vector field F to determine if it is conservative. - $\mathbf { F } = x ^ { 4 } y ^ { 4 } z ^ { 4 } i + x ^ { 4 } y ^ { 4 } z ^ { 4 } j + z ^ { 4 } k$

Find the flux of the vector field F across the surface S in the indicated direction. - $\mathbf { F } = x \mathbf { i } + y \mathbf { j } + z ^ { 3 } \mathbf { k }$ ; S is portion of the cone $z = 3 \sqrt { x ^ { 2 } + y ^ { 2 } }$ between $z = 3$ and $z = 6$ ; direction is outward

Find the work done by F over the curve in the direction of increasing t. - $\mathbf { F } = x y \mathbf { i } + 3 \mathbf { j } + 8 x \mathbf { k } ; \mathrm { C } : \mathbf { r } ( \mathrm { t } ) = \cos 7 \mathrm { ti } + \sin 7 \mathrm { tj } + \mathrm { t } \mathbf { k } , 0 \leq \mathrm { t } \leq \frac { \pi } { 14 }$

Using Green's Theorem, compute the counterclockwise circulation of F around the closed curve C. -F = xyi + xj; C is the triangle with vertices at (0, 0), (9, 0), and (0, 10)

Test the vector field F to determine if it is conservative. - $F = \left( z e ^ { x + y } - \frac { 1 } { x } \right) i + z e ^ { x + y } j + e ^ { x + y } y _ { k }$

Calculate the flow in the field F along the path C. - $\mathbf { F } = \mathrm { y } ^ { 2 } \mathbf { i } + \mathrm { zj } + \mathrm { xk } ; C \text { is the curve } \mathbf { r } ( \mathrm { t } ) = ( 3 + 3 \mathrm { t } ) \mathbf { i } + 2 \mathrm { tj } - 2 \mathrm { tk } , 0 \leq \mathrm { t } \leq 1$