Exam 16: Vector Calculus

The temperature at the point $( x , y , z )$ in a substance with conductivity $K$ is $u ( x , y , z ) = 6 x ^ { 2 } + 6 y ^ { 2 }$ Find the rate of heat flow inward across the cylindrical $y ^ { 2 } + z ^ { 2 } = 6 , \quad 0 \leq x \leq 5$

Find a vector representation for the surface. The plane that passes through the point $( 2,5,1 )$ and contains the vectors $2 \mathbf { i } + 5 \mathbf { j } - 3 \mathbf { k }$ and $2 \mathbf { i } - 3 \mathbf { j } + 5 \mathbf { k }$ ..

Use Stokes' Theorem to evaluate $\iint _ { S } \operatorname { cur } \mathbf { F } \cdot d \mathbf { S }$ . $\mathbf { F } ( x , y , z ) = 4 x y \mathbf { i } + 5 y z \mathbf { j } + 2 z ^ { 2 } \mathbf { k }$ ; S is the part of the ellipsoid $9 x ^ { 2 } + 9 y ^ { 2 } + 4 z ^ { 2 } = 36$ lying above the xy-plane and oriented with normal pointing upward.

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

Show that F is conservative, and find a function f such that $\mathbf { F } = \nabla f$ , and use the result to evaluate $\int _ { C } \mathbf { F } \cdot d \mathbf { r }$ , where C is any curve from $A \left( x _ { 0 } , y _ { 0 } , z _ { 0 } \right)$ to $B \left( x _ { 1 } , y _ { 1 } , z _ { 1 } \right)$ . $\mathbf { F } ( x , y , z ) = 6 x y \mathbf { i } + \left( 3 x ^ { 2 } + 4 y z ^ { 2 } \right) \mathbf { j } + 4 y ^ { 2 } z \mathbf { k }$ ; $A ( 0,0,0 )$ and $B ( 2 , - 2,0 )$

Find an equation of the tangent plane to the parametric surface represented by r at the specified point. $\mathbf { r } ( u , v ) = u e ^ { v } \mathbf { i } + u v \mathbf { j } + v e ^ { - u } \mathbf { k }$ ; u = ln 5, v = 0

Determine whether F is conservative. If so, find a function f such that $\mathbf { F } = \nabla f$ . $\mathbf { F } ( x , y ) = \left( 4 x ^ { 3 } + 2 y \right) \mathbf { i } + \left( 3 x - 4 y ^ { 3 } \right) \mathbf { j }$

Find the area of the surface S where S is the part of the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 1$ that lies inside the cylinder $x ^ { 2 } - x + y ^ { 2 } = 0$

Use Green's Theorem to evaluate the line integral along the positively oriented closed curve C. $\oint _ { C } e ^ { x ^ { 2 } } d x + \left( \sin y - 4 x ^ { 2 } \right) d y$ , where C is the boundary of the region bounded by the parabolas $y = x ^ { 2 }$ and $x = y ^ { 2 }$ .

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. $\mathbf { F } ( x , y , z ) = 9 x \mathbf { i } + 3 y \mathbf { j } + 6 z \mathbf { k }$ $S \text { is the cube with vertices } ( \pm 1 , \pm 1 , \pm 1 ) \text {. }$

Use Stoke's theorem to evaluate $\int _ { C } \mathbf { F } \cdot d \mathbf { r }$ $\mathbf { F } ( x , y , z ) = 6 y x ^ { 2 } \mathbf { i } + 2 x ^ { 3 } \mathbf { j } + 6 x y \mathbf { k }$ C is the curve of intersection of the hyperbolic paraboloid $z = y ^ { 2 } - x ^ { 2 }$ and the cylinder $x ^ { 2 } + y ^ { 2 } = 25$ oriented counterclockwise as viewed from above.

Let D be a region bounded by a simple closed path C in the xy. Then the coordinates of the centroid $( \bar { x } , \bar { y } ) \text { of } D \text { are } \bar { x } = \frac { 1 } { 2 A } \oint _ { C } x ^ { 2 } d y , \bar { y } = - \frac { 1 } { 2 A } \oint _ { C } y ^ { 2 } d x$ where A is the area of D. Find the centroid of the triangle with vertices (0, 0), ( $4$ , 0) and (0, $8$ ).

Use Stoke's theorem to evaluate $\iint _ { S } \mathbf { F } \cdot d r .$ $\mathbf { F } ( x , y , z ) = 4 z \mathbf { i } + 2 x \mathbf { j } + 6 y \mathbf { k }$ C is the curve of intersection of the plane z = x + 9 and the cylinder $x ^ { 2 } + y ^ { 2 } = 9$

Find the curl of the vector field. $\mathbf { F } ( x , y , z ) = ( x - 3 z ) \mathbf { i } + ( 4 x + y + 5 z ) \mathbf { j } + ( x - 6 y ) \mathbf { k }$

Find a parametric representation for the part of the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 64$ that lies above the cone $z = \sqrt { x ^ { 2 } + y ^ { 2 } }$

Find the work done by the force field F on a particle that moves along the curve C. $\mathbf { F } ( x , y ) = ( 5 x + 4 y ) \mathbf { i } + 5 x y \mathbf { j }$ ; $C : \mathbf { r } ( t ) = 2 t ^ { 2 } \mathbf { i } + t ^ { 2 } \mathbf { j }$ , $0 \leq t \leq 1$

Find the mass of the surface S having the given mass density. S is part of the plane $x + y + 3 z = 3$ in the first octant; the density at a point P on S is equal to the square of the distance between P and the xy-plane.

Evaluate $\iint _ { S } \mathbf { F } \cdot d \mathbf { S }$ , that is, find the flux of F across S. $\mathbf { F } ( x , y , z ) = 4 x \mathbf { i } + 2 y \mathbf { j } + 4 \mathbf { k }$ ; S is the part of the paraboloid $z = x ^ { 2 } + y ^ { 2 }$ between the planes z = 0 and z = 5; n points upward.

Use Green's Theorem to find the work done by the force $\mathbf { F } ( x , y ) = x ( x + 4 y ) \mathbf { i } + 4 x y ^ { 2 } \mathbf { j }$ in moving a particle from the origin along the x-axis to (1, 0) then along the line segment to (0, 1) and then back to the origin along the y-axis.

Find the exact value of $\int _ { C } x e ^ { y z } d s ,$ where C is the line segment from (0, 0, 0) to (1, $4$ , $9$ ).