MATH 332: Vector Analysis Sample Final Exams Set 1

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Ivan Avramidi: MATH 332, (Vector Analysis), Final Exam
MATH 332: Vector Analysis
Sample Final Exams
Set 1
1a). Use tensor notation to simplify ( A × B ) · ( C × D )
1b) Use tensor notation to verify the identity
∇ · ( F × G ) = G · (∇ × F ) − F · (∇ × G )
2. (a) Sketch the curve
x = et cos t,
y = et sin t,
(0 ≤ t ≤ 1) .
z = 0,
(b) Determine the arc length of the curve between t = 0 and t = 1. (c) Reparametrize the curve in terms of arc
length.
3. Let C be the curve given by the equation
R (t) = sin t i + cos t j − ln cos t k
0≤t≤
π
2
Find: (a) the unit tangent T ; (b) the unit normal N ; (c) the curvature k;
4. Let F = xy i + y 2 j + z 2 k . (a) Find the general equation of a flow line. (b) Find the flow line through the
point (1, 2, −2).
5. Given the vector field F = (x + xz 2 ) i + xy j + yz k , evalute: (a) div F ; (b) curl F .
6. Let A be a constant vector field, R = x i + y j + z k , and R = | R |. Evaluate
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∇× A ×∇
R
(Hint: Use tensor notation)
7. Let C be the curve given parametrically by
R (t) = 2t i + [t + cos(πt)] j − (t2 + t) k ,
(0 ≤ t ≤ 1)
and F be the vector field
F = (eyz + y − z) i + (xzeyz + x) j + (xyeyz − x) k .
Compute the line integral
R
C
F · dR.
Ivan Avramidi: MATH 332, (Vector Analysis), Final Exam
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8. Let S be the surface given parametrically by
(0 ≤ u ≤ 2,
0 ≤ v ≤ π) ,
p
F = y i − x jRR+ k be a vector RR
field and ϕ(x, y, z) = 1 + x2 + y 2 be a scalar field. Compute the surface
integrals: (a) S F · dS and (b) S ϕ dS.
R = u cos v i + u sin v j + v k ,
9. Let S be the portion of the surface of the sphere x2 + y 2 + z 2 = 9 above the plane z = 1 and below the
plane
z = 2, with normal n pointing upward, and F = (x + yz 3 ) i + xz 3 j − z k be a vector field. Evaluate
RR
F · n dS. (Hint: This problem can be made reasonably simple if you make use of the divergence theorem.)
S
10. Let S be the portion of the surface of the paraboloid z = 9 − x2 − y 2 lying above the plane z = 5, with
normal n pointing upward, and F = (x − yz) i + xz j be a vector field. Verify the Stokes’ theorem.
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Ivan Avramidi: MATH 332, (Vector Analysis), Final Exam
Set 2
1a). Use tensor notation to simplify ( A × B ) × ( C × B )
1b) Use tensor notation to verify the identity
∇ × (∇ × F ) = ∇(∇ · F ) − ∇2 F )
2. a) Describe and sketch the curve
R (t) = t i + sin t j + cos t k .
(b) Determine the arc length, L, of the curve between the points (0, 0, 1) and (2π, 0, 1).
(c) Reparametrize the curve in terms of arc length s.
3. The position vector of a moving particle is
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R (t) = (sin t + cot t) i + (sin t − cos t) j + t k .
2
Find: (a) the velocity, v , (b) the speed, v (c) the acceleration, a (d) the unit tangent T , and the unit normal,
N , to the path of the particle, in the direction of motion, (e) the curvature, k, of the path.
4. Let F be a vector field defined by F = y i − x j + x k . (a) Find the general (parametric) equation of a flow
line. (b) Find the flow line through the point (1, 2, 3).
5. Let R = x i + y j + z k , R = | R |, and F be a vector field defined for R 6= 0 by
F =
R
.
R3
Evalute (a) div F and (b) curl F . (Hint: Notice that F = −∇
1
).
R
6. Let A be a constant vector field, R = x i + y j + z k , and R = | R |. By using the tensor notation show
that for R 6= 0
A
R
∇ × ( A × ∇R) =
+ (A · R)
.
R
R
H
7. Let F = y i +x j +xyz 2 k . Find the integral C F ·d R around the circumference of the circle x2 −2x+y 2 = 2,
z = 1, oriented counterclockwise. (Hint: To parametrize the circle find its center and radius.)
8. Let F be a vector field defined by F = i + xy j , ϕ be a scalar field defined by
−1/2
ϕ(x, y, z) = 1 + x2 + 2xy + y 2 − 2z
, and S be a surface defined parametrically by
R (u, v) = (u + v) i + (u − v) j + u2 k ,
(0 ≤ u ≤ 1,
0 ≤ v ≤ 1) ,
RR
RR
with the normal n pointing upward. Compute the surface integrals: (a) S F · dS and (b) S ϕ dS. (Hint:
The integrals over u and v are very simple!)
9. Let F be a vector field defined by F = xy 2 i + yx2 j + z 2 k , and S be the complete surface of the region
bounded by the cylinder x2 + y 2 = 4 and by the planes z = 0 and z = 2. Use the divergence theorem to evaluate
RR
F · n dS. (Hint: Use cylindrical cordinates to evaluate the volume integral.)
S
10. Let F be a vector field defined by F = 2y i + (x − 2x3 z) j + xy 3 k , and S be a curved surface of the
2
2
2
hemisphere
RR x + y + z = 1, z ≥ 0, with the normal n pointing upward. Use Stokes’ theorem to evaluate the
integral S (∇ × F ) · n dS.
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Ivan Avramidi: MATH 332, (Vector Analysis), Final Exam
Set 3
1. Use tensor notation to simplify {[( B × A ) × A ] × A } · C
2. A curve is given by
R(t) = et cos t i + et sin t j .
Z t
dR(t) dR(τ ) ds
, (b) find s(t) =
= (a) find
dτ dτ , (c) find t(s), (d) reparemetrize the curve in terms of
dt
dt 0
the arc length, s, by substituting t(s) in the equation of the curve, (e) find the values of t corresponding to the
points P1 (1, 0, 0) and P2 (−eπ , 0, 0), (f) find the length of the curve, L, between the points P1 and P2 by using
the arc length function s(t).
3. A particle moves so that its position R at time t is given by
R(t) = e−t cos t i + e−t sin t j + e−t k
Find: (a) the velocity, v , (b) the speed, | v |, (c) the unit tangent, T , (d) the unit normal, N , and (e) the
curvature, k, of the path.
4. Let F be a vector field defined by F =
x2
i + y j + k . Find the general (parametric) equation of a flow
y
line.
5. Let F be a vector field defined by F = x2 y i + z j − (x + y − z) k . Find: (a) div F , (b) curl F , and (c)
grad div F .
6. Let A be a constant vector field and ϕ be a scalar field defined by ϕ = 1 + x + y + z + xyz. By using the
tensor notation show that
curl ( A × grad ϕ) + grad ( A · grad ϕ) = 0 .
(Hint: This equation holds for any harmonic scalar field, i.e. a scalar field satisfying the Laplace equation
∆ϕ = 0, so the particular form of ϕ is not important!)
7. Let F = (3x + 4y) i + (2x + 3y 2 ) j + (exyz ) kI and C be the counterclockwise oriented circle of radius 2 in
F · d R around C. (Hint: The equations of the circle are:
the xy plane with the center at the origin. Find
2
C
2
x + y = 4, z = 0. Parametrize the circle by R (t) = 2 cos t i + 2 sin t j , 0 ≤ t ≤ 2π.)
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8. Let F be a vector field defined by F = x i + y j + (zRR
− 1) k and S be a closed surface bounded by the
2
2
planes z = 0, z = 1 and the cylinder x + y = 1. Find S F · d S . (Hint: The surface S consists of three
parts. Find the normals to each part and compute the surface integrals over each part separately.)
9. Let F be a vector field defined by F = x i − y j , and D be a domain in space bounded by the planes z = 0,
z = 1 and the cylinder x2 + y 2 = 1. Verify the Divergence Theorem, i.e. compute both sides of the divergence
theorem and show that they are equal. (Hint: The boundary of D consists of three parts. Find the normals
to each part compute the surface integrals over each part separately. Use cylindrical coordinates for the volume
integral.)
10. Let F be a vector field defined by F = (y − z) i − (x + z) j + (x + y) k , and S be the portion of the
paraboloid z = 9 − x2 − y 2 that Zlies
Z above the plne z = 0 (with the normal n pointing upward). Use Stokes’
curl F · n dS.
theorem to evaluate the integral
S