Final Exam, Fall 1998, Math 223
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Fall 1998 Final Solutions
Answer the questions in the space provided. You must show your work
or explain your solution; otherwise points may be deducted. If you
make and unnecessary approximation in your solution to a problem, your
answer will be judged on its accuracy. Points may be deducted for
poor or inappropriate approximation.
The points for each problem are as follows: 1 and 2 (12 pts), 3
(6 pts), 4-7 (10 pts each), 8 and 9 (15 pts each).
- 1.
- Label the following statements T (True), F (False),
or C (Can't tell without further information.) No reasons
needed. Assume that
is an arbitrary vector field and
f(x,y,z) is an arbitrary function.
- (a)
- Curl
is a vector field.
- (b)
- Flux of a vector field is a vector.
- (c)
- grad f is parallel to the level surface
f(x,y,z) = c.
- (d)
-
whenever C is a closed
curve.
- (e)
-
is the zero vector
for any vectors
and .
- (f)
- div (grad f) is a vector.
- (g)
-
if C is a
closed curve.
- (h)
- The gradient of a scalar function is a scalar.
- (i)
- A line integral is a scalar.
- (j)
- A directional derivative is a vector.
- (k)
-
is a unit vector if
.
- (l)
-
is a vector.
- 2.
- No partial credit, but you must show your work. Put your answer
in the space provided.
- (a)
- If
and
find
.
- (b)
- If
f(x,y) = exy find grad f.
- (c)
- If
(in three dimensions) find div .
- (d)
- If
(in three dimensions) find curl .
- 3.
- According to the Theory of Relativity, the energy, E, of a
body of mass m moving with speed v is given by the formula
The speed, v, is non-negative and less than the speed of light, c,
which is a constant.
- (a)
- Find
.
- (b)
- Explain what you would expect the sign of
to be and why.
- 4.
- Write a triple integral representing the volume above the cone
and below the sphere of radius 2 centered
at the origin. Include limits of integration, but do not
evaluate:
- (a)
- Use cylindrical coordinates.
- (b)
- Use spherical coordinates.
- 5.
- The sketch shows level curves of the function f(x,y).
- (a)
- On the diagram above, sketch a vector at P in the direction
of grad f.
- (b)
- The length of the vector grad f at P is
longer than | shorter than |
the same length as |
the length of grad f at Q? (Circle one) Give a reason.
- (c)
- If C is a curve going from P to Q, evaluate
Show your reasoning.
- 6.
- The vector field, ,
sketched above, has no
z-component and is independent of z.
- (a)
- Do you think that
is a conservative (that is, path
independent) vector field? Justify your answer using the
sketch.
- (b)
- Do you think that curl
is positive,
negativeor zero? Explain.
- (c)
- Give a possible formula for .
- 7.
- Curves C1 and C2 are parameterized as follows
C1 is
(x(t), y(t)) = (0,t) for
C2 is
for
- (a)
- Sketch C1 and C2 on the axes below, with arrows showing
their orientation. Label your sketch clearly.
- (b)
- Suppose
.
Calculate
,
where C is the curve given by
C = C1 + C2. Show your work.
- 8.
- Consider the vector fields
and
.
Evaluate the
following. No partial credit, but you must show your reasoning.
- (a)
-
where C1 is the unit
circle in the xy-plane, orineted counterclockwise.
- (b)
-
where C2 is the
y-axis from the origin to the point (0,10,0).
- (c)
-
where S1 is the
cylinder
x2 + y2 = 1 with
,
oriented
outward.
- (d)
-
where S2 is the unit
sphere oriented outward.
- 9.
- Let
and let
be the vector field given by
- (a)
- Calculate the flux of
out of the unit sphere
x2 +
y2 + z2 = 1 oriented outward. Show your work.
- (b)
- Calculate div .
Show your work and simplify your
answer completely.
- (c)
- Use your answers to part (a) and (b) to calculate the flux of
out of a box of side 10 centered at the origin with
sides parallel to the coordinate planes. (The box is also
oriented outward.) Give reasons for your answer.
Vector Calculus
1999-04-26