Exam 16: Vector Calculus

Use Stoke's theorem to calculate the surface integral $\iint _ { S } \mathbf { F } \cdot d \mathbf { S } ,$ where $\mathbf { F } ( x , y , z ) = \left( e ^ { x } \sin y \right) \mathbf { i } + \left( e ^ { x } \cos y - z \right) \mathbf { j } + y \mathbf { k }$ and S is the part of the cone $z ^ { 2 } = x ^ { 2 } + y ^ { 2 } + 3 \text { for which } 1 \leq z \leq 2$

Find parametric equations for C, if C is the curve of intersection of the hyperbolic paraboloid $z = y ^ { 2 } - x ^ { 2 }$ and the cylinder $x ^ { 2 } + y ^ { 2 } = 4$ oriented counterclockwise as viewed from above.

Determine whether or not F is a conservative vector field. If it is, find a function f such that $\mathbf { F } = \nabla f .$ $\mathbf { F } = ( 6 x \cos y - y \cos x ) \mathbf { i } + \left( - 3 x ^ { 2 } \sin y - \sin x \right) \mathbf { j }$

Evaluate the line integral \cdotd, where $\mathbf { F } ( x , y ) = ( x - y ) \mathbf { i } + ( x y ) \mathbf { j }$ and C is the arc of the circle $x ^ { 2 } + y ^ { 2 } = 25$ traversed counterclockwise from ( $5$ , 0) to (0, - $5$ ). Round your answer to two decimal places.

Find the area of the surface S where S is the part of the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 16$ that lies to the right of the xz-plane and inside the cylinder $x ^ { 2 } + z ^ { 2 } = 9$

Let S be the cube with vertices $( \pm 1 , \pm 1 , \pm 1 )$ . Approximate $\iint _ { S } \sqrt { x ^ { 2 } + 2 y ^ { 2 } + 7 z ^ { 2 } }$ by using a Riemann sum as in Definition 1, taking the patches $S _ { i j }$ to be the squares that are the faces of the cube and the points $P _ { i j }$ to be the centers of the squares.

Find a parametric representation for the part of the plane $z = 2$ that lies inside the cylinder $x ^ { 2 } + y ^ { 2 } = 9$

Find the area of the part of the cone $z = \sqrt { x ^ { 2 } + y ^ { 2 } }$ that is cut off by the cylinder $( x - 3 ) ^ { 2 } + y ^ { 2 } = 4$

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

Evaluate the line integral over the given curve C. $\int _ { C } 4 x y d s$ , where C is the line segment joining (-2, -1) to (4, 5)

Use the Divergence Theorem to calculate the surface integral $\iint _ { 5 } \mathbf { F } \cdot d \mathbf { S }$ ; that is, calculate the flux of $F$ across $S$ . $\mathbf { F } ( x , y , z ) = x y e ^ { x } \mathbf { i } + x y ^ { 2 } z ^ { 3 } \mathbf { j } - y e ^ { x } \mathbf { k }$ S is the surface of the box bounded by the coordinate planes and the planes $x = 4 , y = 2 \text { and } z = 1$ .

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 $B \left( x _ { 1 } , y _ { 1 } , z _ { 1 } \right)$ . $\mathbf { F } ( x , y , z ) = 12 x ^ { 2 } y \mathbf { i } + \left( 4 x ^ { 3 } + 10 y z ^ { 2 } \right) \mathbf { j } + 10 y ^ { 2 } z \mathbf { k }$ ; $A ( 0,0,0 )$ and $B ( 0,3,1 )$

Find the divergence of the vector field. $\mathbf { F } ( x , y , z ) = \frac { x } { 4 x ^ { 2 } + 7 y ^ { 2 } + 3 z ^ { 2 } } \mathbf { i } + \frac { y } { 4 x ^ { 2 } + 7 y ^ { 2 } + 3 z ^ { 2 } } \mathbf { j } + \frac { z } { 4 x ^ { 2 } + 7 y ^ { 2 } + 3 z ^ { 2 } } \mathbf { k }$

Find the exact value of $\int _ { C } \mathbf { F } \cdot d \mathbf { r }$ where $\mathbf { F } ( x , y ) = e ^ { x - 1 } \mathbf { i } + x y \mathbf { j }$ and C is given by $\mathbf { r } ( t ) = 2 t ^ { 2 } \mathbf { i } + 7 t ^ { 3 } \mathbf { j } , 0 \leq t \leq 1$

Match the vector field with one of the plots shown below. $\mathbf { F } ( x , y , z ) = \mathbf { j } + \mathbf { k }$

Find the area of the surface. The part of the plane $\mathbf { r } ( u , v ) = ( 5 u + 3 v + 4 ) \mathbf { i } + ( 2 u + 3 v + 4 ) \mathbf { j } + ( 4 u + 2 v + 8 ) \mathbf { k }$ ; $0 \leq u \leq 1$ , $0 \leq v \leq 5$

Let $\mathbf { r } = x \mathbf { i } + y \mathbf { j } + z \mathbf { k } \text { and } r = | \mathbf { r } | \text {. }$ $\text { Find } \nabla \cdot ( 9 \mathbf { r } )$

Assuming that S satisfies the conditions of the Divergence Theorem and the scalar functions and components of the vector fields have continuous second order partial derivatives, find $\iint _ { 5 } 3 \mathbf { a } \mathbf { n } d S$ , where a is the constant vector.

Evaluate the line integral over the given curve C. $\int _ { C } 4 x y d s$ , where C is the line segment joining (-4, -5) to (5, 4)

Use Green's Theorem to find the work done by the force $\mathbf { F } ( x , y ) = \left( 6 x - 7 y ^ { 2 } \right) \mathbf { i } + 3 y \mathbf { j }$ in moving a particle in the positive direction once around the triangle with vertices $( 0,0 )$ , $( 1,0 )$ , and $( 0,1 )$ .