vector calculus1

 

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Math 223

Disclaimer:
It is not a good idea to rely exclusively on reading through old exam solutions as a
way to prepare for the final exam. In particular, this semester’s course director
may not have written any of the exams available from this page, so the ones he/she
gives will almost certainly have a somewhat different flavor.

Topic Warning:
Because the topics taught differ slightly from semester to semester, it is not a good
idea to use the old exams to gauge the content of the exams this semester.

Math 223 Show Your Work! Page 1 of 1

1

1. (8) Consider two vectors

#„v = 4
#„
i �

#„
j + a

#„
k

and #„w = a

#„
i + 5

#„
j �

#„
k . For what values of

a are #„v and #„w perpendicular?

2. (8) Find an equation for the plane passing through the points

(0, 2, 1), (1, 1, 5), and (2, 0, 11).

3. (8) Find a vector of magnitude 10 normal to the plane 5x + 3y = 6z + 1.

Math 223 Show Your Work! Page 2 of 11

4. (8) At which point (or points) on the ellipsoid x2 + 4y2 + z2 = 9 is the tangent plane
parallel to the plane z = 0?

5. (8) Find a parametric equation for a line through the point (1, �3, 5) and parallel to the
vector 5

#„
i + 3

#„
j �
#„
k

Math 223 Show Your Work! Page 3 of 11

6. (8) Find the directional derivative of f(x, y) = x2y + y2x at the point (1, �1) in the
direction of 3

#„
i � 4

#„
j .

7. (7) Compute the flux integral

Z

S

#„
F ·

# „
dA where

#„
F =

#„
i +

#„
j �

#„
k and S is the surface

z = x2 � y2, 0  x  3, 0  y  3, oriented upwards.

Math 223 Show Your Work! Page 4 of 11

8. (20) Consider the integral

Z 1

0

Z

2

(8y)1/3

1

1 + x4
dxdy.

(a) Interchange the order of integration. Show your work by including a sketch of the

region of integration.

(b) Evaluate the integral.

Math 223 Show Your Work! Page 5 of 11

9. (20) Let

#„
F =

x

#„
i +

y

#„
j + z

#„
k . Evaluate the following:

(a)

Z

C

#„
F ·

#„
dr where C is the line from (0, 0, 0) to (1, 1, 1).

(b)

Z
S
#„
F ·

# „
dA where S is the triangle in the plane y = 10 with vertices (0, 10, 0), (4, 10, 0),

and (0, 10, 1), oriented in the direction of increasing y.

(c)

Z
S
#„
F ·

# „
dA where S is the sphere of radius 2 centered at (5, 5, 0), oriented outward.

Math 223 Show Your Work! Page 6 of 11

10. (15) Let H(x, y, z) = sin (2x + y) + z. Find the equation of the tangent plane to the
level surface H(x, y, z) = 5 at the point (⇡, ⇡, 5).

Math 223 Show Your Work! Page 7 of 11

11. (20) Let

#„
F = (y + z)x

#„
i + y

#„
j + xyz

#„
k .

(a) Find

curl(

#„
F ).

(b) Let S be the surface x2 + y2 + z = 25, with 0  z  25, oriented upward. Find the
value of the flux integral

Z
S
curl(

#„
F ) ·

# „
dA.

Math 223 Show Your Work! Page 8 of 11

12. (15) Let a be a constant, a 6= 2, and consider function f(x, y) =
1

2

x2 + 2y + 2xy + ay2.

(a) Find the critical point of f.

(b) Find all values of a so that the critical point is a global minimum.

Math 223 Show Your Work! Page 9 of 11

13. (20) Consider the contour diagram for the function f(x, y) sketched below.

0

0.015625

0.125

0.421875

1

1.95313

3.375

5.35938

8
-2 -1 1

x

-2

-1

1
2
y

(a) Sketch a graph of f(x, 0).

(b) Determine whether the following quantities are positive, negative, or equal to zero.

f
xx

(0, 0) is f
xy

(0, 0) is

(c) If all contour lines are parallel to the line 2x + y = 0, then determine the direction in
which the gradient of f points, as a unit vector.

Math 223 Show Your Work! Page 10 of 11

14. (15) Rewrite the integral

Z 3

�3

Z 0

�
p

9�y2

Z p18�x2�y2
p

x

2+y2
xy dz dx dy in spherical and cylin-

drical coordinates.

(a) In spherical coordinates, use the order of integration d⇢ d✓ d�.

(b) In cylindrical coordinates, use the order of integration dz dr d✓.

Math 223 Show Your Work! Page 11 of 11

15. (20) Consider the 2-dimensional force field

#„
F = 2xex

2�5y #„i � 5ex2�5y
#„
j .

(a) Is

#„
F conservative? If so, find a potential function f(x, y) whose gradient is

#„
F .

(b) Find the work done by the force field

#„
F in moving an object from P(0, 2) to Q(�2, 0)

along the path formed by C1 followed by C2 as shown in the figure below. C1 and C2
may be parametrized as follows:

C1 : x = t, y = 2 � t, 0  t  2,
C2 : x = 2 cos t, y = �2 sin t, 0  t  ⇡ .

• Math 223
• Fall12_223