Exam 16: Vector Calculus

Evaluate $\iint _ { S } f ( x , y , z ) d S$ . $f ( x , y , z ) = x + y$ ; S is the part of the plane $7 x + 3 y + z = 21$ in the first octant.

Use Green's Theorem to evaluate the line integral along the positively oriented closed curve C. $\oint _ { C } 3 x y d x + 4 x ^ { 2 } d y$ , where C is the triangle with vertices $( 0,0 )$ , $( 3,4 )$ , and $( 0,4 )$ .

Find the gradient vector field of $f ( x , y ) = \ln ( x + 2 y )$

Use Stokes' Theorem to evaluate $\iint _ { S } \operatorname { curl } \mathbf { F } \cdot d \mathbf { S }$ S consists of the top and the four sides (but not the bottom) of the cube with vertices $( \pm 9 , \pm 9 , \pm 9 )$ oriented outward. $\mathbf { F } ( x , y , z ) = 4 x y z \mathbf { i } + 4 x y \mathbf { j } + 4 x ^ { 2 } y z \mathbf { k }$

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

Evaluate the surface integral where S is the surface with parametric equations $x = 7 u v$ , $y = 6 ( u + v ) , z = 6 ( u - v ) , u ^ { 2 } + v ^ { 2 } = 3$ . $\iint _ { S } 10 y z d S$

Evaluate $\iint _ { S } f ( x , y , z ) d S$ . $f ( x , y , z ) = z$ ; S is the part of the torus with vector representation $\mathbf { r } ( u , v ) = ( 5 + 3 \cos v ) \cos u \mathbf { i } + ( 5 + 3 \cos v ) \sin u \mathbf { j } + 3 \sin v \mathbf { k }$ , $0 \leq u \leq 2 \pi$ , $0 \leq v \leq \frac { \pi } { 2 }$ .

Suppose that $f ( x , y , z ) = g \left( \sqrt { x ^ { 2 } + y ^ { 2 } + z ^ { 2 } } \right)$ where g is a function of one variable such that $g ( 5 ) = 8$ . Evaluate $\iint _ { S } f ( x , y , z ) d S$ where S is the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 25$

Find the mass of the surface S having the given mass density. S is the hemisphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 9$ , $z \geq 0$ ; the density at a point P on S is equal to the distance between P and the xy-plane.

Show that F is conservative and find a function f such that $\mathbf { F } = \nabla f$ , and use this result to evaluate $\int _ { C } \mathbf { F } \cdot d \mathbf { r }$ , where C is any path from $A \left( x _ { 0 } , y _ { 0 } \right)$ to $B \left( x _ { 1 } , y _ { 1 } \right)$ . $\mathbf { F } ( x , y ) = \left( 12 x ^ { 2 } y ^ { 2 } - 16 x y ^ { 3 } \right) \mathbf { i } + \left( 8 x ^ { 3 } y - 24 x ^ { 2 } y ^ { 2 } \right) \mathbf { j }$ ; $A ( 3,1 )$ and $B ( 3,2 )$

Evaluate the line integral over the given curve C. $\int _ { C } \left( 3 x + 7 y ^ { 3 } \right) d s$ ; $C : \mathbf { r } ( t ) = ( t - 3 ) \mathbf { i } + t \mathbf { j }$ , $0 \leq t \leq 3$

Evaluate the line integral. 4yzxds C:x=t,y=t,z=t0\leqt\leq\pi

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. $\nabla ( \operatorname { grad } f )$

Evaluate $\int _ { C } y z d y + x y d z$ , where C is given by $x = 10 \sqrt { t } , y = 3 t , z = 10 t ^ { 2 } , 0 \leq t \leq 1$ Round your answer to two decimal place.

Find the area of the surface S where S is the part of the plane $z = 2 x ^ { 2 } + y$ that lies above the triangular region with vertices $( 0,0 ) \text {, }$ $( 3,0 )$ , and $( 3,3 )$

Use Stokes' Theorem to evaluate $\iint _ { S } \operatorname { cur } \mathbf { F } \cdot d \mathbf { S }$ . $\mathbf { F } ( x , y , z ) = 6 x y \mathbf { i } + 5 y z \mathbf { j } + 2 z ^ { 2 } \mathbf { k }$ ; S is the part of the paraboloid $z = x ^ { 2 } + y ^ { 2 }$ lying below the plane $z = 6$ and oriented with normal pointing downward.

Evaluate the line integral over the given curve C. $\int _ { C } \left( 8 x + 2 y ^ { 2 } \right) d s$ ; $C : \mathbf { r } ( t ) = ( t - 8 ) \mathbf { i } + t \mathbf { j }$ , $0 \leq t \leq 3$

Use Stoke's theorem to evaluate $\iint _ { S } \mathbf { F } \cdot d r .$ $\mathbf { F } ( x , y , z ) = 2.56 x \mathbf { i } + 16 y \mathbf { j } + 4 \left( y ^ { 2 } + x ^ { 2 } \right) \mathbf { k }$ C is the boundary of the part of the paraboloid $z = 2.56 - x ^ { 2 } - y ^ { 2 }$ in the first octant. C is oriented counterclockwise as viewed from above.

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

Determine whether or not F is a conservative vector field. If it is, find a function f such that $\mathbf { F } = \nabla f .$ $\mathbf { F } = ( 4 + 2 x y + \ln x ) \mathbf { i } + x ^ { 2 } \mathbf { j }$