Math 227-103 (CRN 10968)
Carter
2015
Final Exam, Dec. 9
Fall
General Instructions. Write your name on only the outside of your blue books. Do not
write on this test sheet, do all of your work inside your blue books. Write neat complete
solutions to each of the problems in the blue book. Please put your test sheet inside the blue
book as you leave. There are 160 points. For a quick dinner, try frozen ravioli with some
red sauce.
Thank you for your efforts in this course. I enjoyed having each of you in my class.
Suppose that C is a closed curve that bounds an orientable region A with
compatible orientation; suppose that G is a closed orientable surface that
bounds a region Ω in space; suppose that F is a vector field in space such
that all of its derivatives are defined in space. Recall that:
•
I
Z Z
F · dr =
(∇ × F ) · n dσ,
C
A
and
•
Z Z
Z Z Z
F · n dσ =
G
(∇ · F ) dV,
Ω
where
∇=
∂
∂
∂
i+ j+ k
∂x
∂y
∂z
1
1. (20 points) Evaluate the integral
Z (3,5,7)
(yz) dx + (xz) dy + (xy) dz.
(2,3,5)
2. Consider the function
f (x, y) = (x − 2)2 − (y − 1)2 + 5
over the triangular region such as that bounded by y = 0, x = 0 and
x + y = 4.
(a) (5 points) Compute the values of f at the points (0, 0), (0, 4), and
(4, 0).
(b) (5 points) Determine the critical points and critical values of the
restriction f (x, 0), where x ∈ [0, 4].
(c) (5 points) Determine the critical points and critical values of the
restriction f (0, y), where y ∈ [0, 4].
(d) (5 points) Determine the critical points and critical values of the
restriction of z = f (x, y) on the line segment x + y = 4 when x ∈
[0, 4].
(e) (5 points) Determine the critical point(s) of z = f (x, y). Classify as
local Max., local min. or saddle points. Compute the critical value.
(f) (5 points) What are the Maximal and minimal values of z = f (x, y)
and where do they occur?
3. (20 points) Use Green’s Theorem to evaluate
I
(x − y) dx + (y − x) dy
C
where C is the unit circle {(x, y) : x2 + y 2 = 1}.
4. (20 points) Compute the gradient vector field for the function
p
f (x, y, z) = ln x2 + y 2 + z 2 .
2
5. (20 points) Compute
1
2
Z
(x dy − y dx)
C
for the curve
r(t) = R cos (t)i + R sin (t)j
for t ∈ [0, 2π]. Use Green’s Theorem to explain the geometric meaning
of your answer.
6. Consider the sphere of radius A that is given by the parametric equations:
S(φ, θ) = A sin (φ) cos (θ)i
+A sin (φ) sin (θ)j + A cos (φ)k.
(a) (5 points) Compute the vector Sφ whose components consist of the
partials of the component functions with respect to the variable φ.
(b) (5 points) Compute the vector Sθ whose components consist of the
partials of the component functions with respect to the variable θ.
(c) (5 points) Compute the cross-product
Sφ × Sθ .
(d) (5 points) Set up the integral that is used to compute the area of
this sphere.
(e) (5 points) Compute the surface area.
3
7. Consider the vector field
F =
where ρ =
p
xi + yj + zk
ρ3
x2 + y 2 + z 2 .
(a) (5 points) Compute
∂ρ
∂x .
(b) (5 points) Use symmetry to give the values for
∂ρ
∂y
and
∂ρ
∂z .
∂
(c) (5 points) Use the results above to compute ∂x
(xρ−3 ), and again
∂
∂
symmetrically compute ∂y
(yρ−3 ) and ∂z
(zρ−3 ).
(d) (5 points) Use the divergence theorem to compute
Z Z
F · n dσ
G
where G is the boundary of the domain Ω = {(x, y, z) : a2 ≤ x2 +
y 2 + z 2 ≤ b2 } n is the outward pointing normal, and dσ denotes the
surface area form.
(e) (5 points) Now compute the outward flux of F across the surface of
a ball of radius a that is centered at the origin. Specifically, compute
Z Z
F · n dσ
S
where
S = {(x, y, z) : x2 + y 2 + z 2 = A2 }
whose outward pointing unit normal is n =
4
xi+yj+zk
.
A