Exam 17: Parameterization and Vector Fields

Let f(x, y)be a function that depends on only one of the variables, that is, of the form f(x, y)= g(x)or f(x, y)= g(y). Could the following picture be the gradient of f?

Write a formula for a vector field $\vec { F } ( x , y )$ whose vectors are parallel to the x-axis and point away from the y-axis, with magnitude inversely proportional to the cube of the distance from the x-axis.

Find parametric equations for the sphere $( x - 6 ) ^ { 2 } + ( y + 6 ) ^ { 2 } + ( z - 12 ) ^ { 2 } = 36$

Parameterize the curve which lies on the plane 5x - 10y + z = 6 above the circle $x ^ { 2 } + y ^ { 2 } = 25$

Find a parametric equation for the line which passes through the point (5, 1, -1)and is parallel to the line $( 2 + 2 t ) \vec { i } + ( 3 - t ) \vec { j } - ( 3 + 2 t ) \vec { k }$ .

Give parameterizations for a circle of radius 2 in the plane, centered at origin, traversed anticlockwise.

A particle moves with position vector $\vec { r } ( t ) = \ln t \vec { i } + t ^ { - 1 } \vec { j } + e ^ { - t } \vec { k }$ . Describe the movement of the particle as t $\rightarrow$ $\infty$ .

Let S be a circular cylinder of radius 0.2, such that the center of one end is at the origin and the center of the other end is at the point (5, 0, 4). Find the xyz-equation of the plane, P, containing the base of the cylinder (i.e., the plane through the origin perpendicular to the axis of the cylinder).

Find a parameterization of the curve $y ^ { 4 } = x ^ { 5 }$ and use it to calculate the path length of this curve from (0, 0)to (1, 1).

The line through the points (2, 5, 25)and (12, 7, 23)can be parameterized by $\vec { r } = 2 \vec { i } + 5 \vec { j } + 25 \vec { k } + t ( 10 \vec { i } + 2 \vec { j } - 2 \vec { k } )$ . What value of t gives the point (42, 13, 17)?

The path of an object moving in xyz-space is given by $( x ( t ) , y ( t ) , z ( t ) ) = \left( 4 t ^ { 2 } , 2 t + 1,2 t ^ { 3 } \right)$ The temperature at a point (x,y,z)in space is given by $f ( x , y , z ) = x ^ { 2 } y - 2 z$ . (a)At time $t = 1$ , what is the object's velocity $\vec { v }$ ? What is its speed? (b)Calculate the directional derivative of f in the direction of $\vec { v }$ at the point (1,2,1), where $\vec { v }$ is the velocity vector you found in part (a). (c)Calculate $\left. \frac { d } { d t } f ( x ( t ) , y ( t ) , z ( t ) ) \right| _ { t - 1 }$ (d)Explain briefly how your answers to part (a), (b)and (c)are related.Interpret them in terms of temperature.

Consider the curve $\vec { r } ( t ) = ( t + 1 ) \vec { i } + ( 2 - t ) \vec { j } + \left( 2 t ^ { 2 } - t + 3 \right) \vec { k } , 0 \leq t \leq 1$ . (a)Find a unit vector tangent to the curve at the point (1,2,3). (b)Show that the curve lies on the surface $z = x ^ { 2 } + y ^ { 2 } - y$ .

In an exam, students are asked to find the arc length of the curve C parameterized by $\vec { r } ( t ) = \left( \frac { t ^ { 2 } } { 2 } - \frac { t ^ { 4 } } { 4 } \right) \vec { i } + \frac { 2 t ^ { 3 } } { 3 } \vec { j } + \vec { k }$ , for $- 1 \leq t \leq 1$ . One student wrote the following " $\| \vec { v } ( t ) \| = \sqrt { \left( t - t ^ { 3 } \right) ^ { 2 } + \left( 2 t ^ { 2 } \right) ^ { 2 } } = \sqrt { t ^ { 2 } \left( 1 + 2 t + t ^ { 4 } \right) } = \sqrt { t ^ { 2 } \left( 1 + t ^ { 2 } \right) ^ { 2 } } = t \left( 1 + t ^ { 2 } \right)$ Thus the arc length is $\int _ { - 1 } ^ { 1 } t \left( 1 + t ^ { 2 } \right) d t = 0$ " This answer cannot be true. (a)Which part of the student's calculation was wrong? (b)Find the correct answer.

Answer the following as "true", "false" or "need more information". If a particle moves with velocity $\vec { v } = 2 t \vec { i } + 3 t \vec { j } + 4 t \vec { k }$ , then the particle stops at the origin.

Consider the parametric surface $\vec { r } ( s , t ) = s \sin \left( \frac { \pi } { 2 } \right) \vec { i } + s \cos \left( \frac { \pi } { 2 } \right) \vec { j } + 2 t \vec { k }$ Does it contain the point (0, -3, -4)?

What curve, C, is traced out by the parameterization $\vec { r } ( t ) = 2 \vec { i } + 4 \cos t \vec { j } + 4 \sin t \vec { k }$ for $0 \leq t \leq 2 \pi$ ? Give a very complete verbal description.

Find the parametric equations for the line of the intersection of the planes $4 x + y + 4 z = 7$ and $x + 4 y + 4 z = 1$ .

Which of the following equations give alternate parameterizations of the line L parameterized by $\vec { r } = ( 1 + 2 t ) \vec { i } + ( 2 + 2 t ) \vec { j } - ( 1 + 4 t ) \vec { k } ?$

If the flow lines for the vector field $\vec { F } ( \vec { r } )$ are all concentric circles centered at the origin, then $\vec { H } ( \vec { r } ) \cdot \vec { r } = 0$ for all $\vec { r }$

Suppose $\vec { F } ( x , y , z ) = - 6 x ^ { 2 } \vec { i } + 12 y ^ { 3 } \vec { j }$ Find a function f(x, y, z) of three variables with the property that the vectors in $\vec { F }$ on a level surface of f (x, y, z) are perpendicular to the level surface of f(x, y, z) at each point.