Exam 16: Vector Calculus

Find a parametric representation for the part of the elliptic paraboloid $x + y ^ { 2 } + 2 z ^ { 2 } = 7$ that lies in front of the plane x = 0.

Let F be a vector field. Determine whether the expression is meaningful. If so, state whether the expression represents a scalar field or a vector field. curl (div F)

Evaluate $\int _ { C } x y ^ { 4 } d S$ where C is the right half of the circle $x ^ { 2 } + y ^ { 2 } = 9$

A thin wire is bent into the shape of a semicircle $x ^ { 2 } + y ^ { 2 } = 4 , x > 0$ If the linear density is $4$ , find the exact mass of the wire.

Find the gradient vector field of $f ( x , y , z ) = x \cos \frac { 2 y } { 9 z }$

Match the vector field with its plot. $\mathbf { F } ( x , y ) = \frac { x } { x ^ { 2 } + y ^ { 2 } } \mathbf { i } - \frac { y } { x ^ { 2 } + y ^ { 2 } } \mathbf { j }$

Find the area of the surface S where S is the part of the surface $x = y z$ that lies inside the cylinder $y ^ { 2 } + z ^ { 2 } = 16$

Find an equation in rectangular coordinates, and then identify the surface. $r ( u , v ) = 6 v \mathbf { i } + ( 8 u - v ) \mathbf { j } + ( u + 6 v ) \mathbf { k }$

Find a function f such that $\mathbf { F } = \nabla f$ , and use it to evaluate $\int _ { C } \mathbf { F } \cdot d \mathbf { r }$ along the given curve C. $\mathbf { F } ( x , y ) = e ^ { 6 y } \mathbf { i } + \left( 1 + 6 x e ^ { 6 y } \right) \mathbf { j } , \quad C : \mathbf { r } ( t ) = t e ^ { t } \mathbf { i } + ( 1 + t ) \mathbf { j } , 0 \leq t \leq 1$

Let $\mathbf { r } = x \mathbf { i } + y \mathbf { j } + z \mathbf { k } \text { and } r = | \mathbf { r } | \text {. }$ $\text { Find } \nabla \cdot ( 8 r \mathbf { r } )$

The plot of a vector field is shown below. A particle is moved $\text { from the point }$ $( 3,3 )$ $\text { to }$ $( 0,0 )$ . By inspection, determine whether the work done by F on the particle is positive, negative, or zero.

Evaluate the surface integral. Round your answer to four decimal places. $\iint _ { S } 3 z d S$ S is surface $x = y ^ { 2 } + 2 z ^ { 2 } , 0 \leq y \leq 1,0 \leq z \leq 1$

Evaluate the line integral over the given curve C. $\int _ { C } 3 y ^ { 2 } z d s$ ; $C : \mathbf { r } ( t ) = 10 t \mathbf { i } + \sin 7 t \mathbf { j } + \cos 7 t \mathbf { k }$ , $0 \leq t \leq \frac { \pi } { 2 }$

Find the work done by the force field $\mathbf { F } ( x , y ) = x \sin ( y ) \mathbf { i } + y \mathbf { j }$ on a particle that moves along the parabola $y = x ^ { 2 } \text { from } ( 1,1 ) \text { to } ( 2,4 )$

Determine whether or not vector field is conservative. If it is conservative, find a function f such that $\mathbf { F } = \nabla f .$ $\mathbf { F } ( x , y , z ) = 35 y z e ^ { 5 x z } \mathbf { i } + 7 e ^ { 5 x z } \mathbf { j } + 35 x y e ^ { 5 x z } \mathbf { k }$

Evaluate the surface integral $\iint _ { S } \mathbf { F } \cdot d \mathbf { S }$ for the given vector field F and the oriented surface S. In other words, find the flux of F across S. $\mathbf { F } ( x , y , z ) = x \mathbf { i } - z \mathbf { j } + y \mathbf { k } , \text { Sis the sphere } x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 25$ in the first octant, with orientation toward the origin.

Determine whether F is conservative. If so, find a function f such that $\mathbf { F } = \nabla f .$ . $\mathbf { F } ( x , y , z ) = ( 6 \sinh 2 z ) \mathbf { i } + \left( 3 e ^ { 5 z } \cos 3 y \right) \mathbf { j } + ( 12 x \cosh 2 z ) \mathbf { k }$