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Final Exam Review Pack
1 Lecture 23: Vector Fields
Summary of Lecture
1. A vector field is a vector function of position. That is, a vector field is a function from R2 (2 dimensional)or R3 (3 dimensional) that assigns each point a vector 〈f(x, y), g(x, y)〉 (in 3 dimensions it takes the form〈f(x, y, z), g(x, y, z), h(x, y, z)〉). We call the functions f, g, h the component functions of the vector field.
2. The domain of the vector field is the intersection of the domains of component functions. A vector field iscontinuous/differentiable if and only if all the components are continuous/differentiable. Vector fields areused in many applications in physics (gravitational, electricity, velocity fields) and other fields. In 2 and 3dimensional setting, vector fields can be visualized as a set of arrows in 2 or 3 space.
3. A radial vector field has the form F = Cr
‖r‖p where p and r are constants. Here r = 〈x, y〉 in 2 dimensions
and r = 〈x, y, z〉 in 3 dimensions.
4. A potential function for a vector field F is a scalar function φ such that F = ∇φ.
5. A vector field F is said to be a conservative vector field if F has a potential function φ
6. A vector field F = 〈F1, F2, F3〉 satisfies the cross partial condition (equivalently, irrotational) if
∂F2
∂x=∂F1
∂y
∂F3
∂y=∂F2
∂z
∂F1
∂z=∂F3
∂x
7. All conservative vector fields satisfy the cross partial condition. But the converse is not true. That is, thereare non conservative vector fields that satisfy the cross partial condition.
8. If the domain of the vector field F is simply connected, then F is conservative if and only if it satisfies thecross partial condition.
1
Exercises
1. Is the vector field F = 〈yz, xz, y〉 conservative?
2. Find a potential function to the vector field F = 〈x, y〉
3. Find a potential function to the vector field F = 〈yexy,xexy〉
4. Find a potential function to the vector field F = 〈yz2, xz2, 2xyz〉
5. Find a potential function to the vector field F = 〈2xzex2
, 0, ex2〉
6. Find a potential function φ(x, y) for F =r
‖r‖3 and a potential function ψ(x, y, z) G =r
‖r‖4 in R3.
7. Which of (A) or (B) in the following figure is the contour plot of a potential function for the vector field F.Recall that the gradient vectors are orthogonal to the level curves.
Answers
1. Not conservative.
2. φ(x, y) =1
2x2 +
1
2y2
3. φ(x, y) = exy
4. φ(x, y, z) = xyz2
5. φ(x, y, z) = zex2
6. φ = − 1
2‖r‖2 and ψ = − 1
3‖r‖3
7. (B)
2
2 Lecture 24: Line integrals of scalar functions and vector fields
Summary of Lecture
1. The scalar line integral of a scalar function f(x, y, z) over a curve C is defined by
∫
C
f(x, y, z) ds = lim∆si→0
N∑
i=1
f(xi, yi, zi)∆si
where s is the arc length.
2. the computational formula for the scalar line integral of a function f(x, y, z) on a smooth curve C which hasthe parametric equation r(t) = 〈x(t), y(t), z(t)〉, t ∈ [α, β] is given by
∫
C
f(x, y, z) ds =
∫ β
α
f(x(t), y(t), z(t))‖r′(t)‖ dt
3. The vector line integral of a vector field F on a smooth curve C is defined by∫CF · T ds. This sums the
component of F with respect to the unit tangent vector T of the curve C. This is also called the circulationof F around C.
4. The computational formula for the vector line integral of a vector field F = 〈F1, F2, F3〉 on a smooth curve Cwhich has the parametric equation r(t) = 〈x(t), y(t), z(t)〉, t ∈ [α, β] is given by
∫
C
F ·T ds =
∫ β
α
F(x(t), y(t), z(t)) · r′(t) dt
5. There are various notations for the vector line integral of a vector field F = 〈f, g, h〉 over a smooth curve C.
∫
C
F ·T ds =
∫ β
α
F(r(t)) · r′(t) dt =
∫ β
α
(fx′(t) + gy′(t) + hz′(t)) dt =
∫
C
f dx+ g dy + h dz =
∫
C
F · dr
6. Properties of vector line integrals: Let C be a smooth oriented curve and F and G are vector fields, then
(a)∫C
(F+G) dr =∫C
(F) dr +∫C
(G) dr (linearity)
(b) If −C denotes C with reverse orientation, then∫−C(F) dr = −
∫C
(F) dr.
(c) If C is a union of piecewise smooth curves C1 and C2, then∫C
(F) dr =∫C1
(F) dr+∫C2
(F) dr. This canbe extended to any number of pieces.
7. If F represents a force field, the work done moving an object from a point a to b along a path r(t) is given by∫ b
a
F ·T ds.
3
Exercises
1. Let f(x, y, z) = x+ yz and let C be the line segment from P = (0, 0, 0) to (6, 2, 2).
(a) Calculate f(c(t)) and ds = ‖c′(t)‖dt for the parametrization c(t) = 〈6t, 2t, 2t〉 for 0 ≤ t ≤ 1.
(b) Evaluate
∫
C
f(x, y, z) ds.
2. Let F = 〈y2, x2〉 and let C be the curve y = x−1 for 1 ≤ x ≤ 2 oriented from left to right.
(a) Calculate F(c(t)) and ds = c′(t) dt for the parametrization of C given by c(t) = 〈t, t−1〉.
(b) Calculate the dot product F(c(t)) · c′(t) and evaluate
∫
C
F · dS.
3. Compute the integral of the scalar function f(x, y, z) = x2 + y2 + z2 on the curve r(t) = 〈cos(t), sin(t), t〉 fort ∈ [0, π]
4. Compute the line integral of the vector field F = 〈x, y, z〉 on the curve r(t) = 〈cos(t), sin(t), t〉 for t ∈ [0, π].
5. Compute the integral of the scalar function f(x, y) =√
1 + 9xy on the curve y = x2 for 0 ≤ x ≤ 1.
6. Compute the line integral
∫
C
f ds of the function f(x, y) =y3
x7on the curve y =
1
4s4 for 1 ≤ x ≤ 2.
7. Compute the line integral
∫
C
f ds of the function f(x, y, z) = xez2
on the piecewise linear path from (0, 0, 1)
to (0, 2, 0) to (1, 1, 1).
8. Compute the line integral
∫
C
f ds of the function f(x, y, z) = x2z on the curve given by r(t) = 〈et,√
2t, e−t〉,0 ≤ t ≤ 1
9. Calculate
∫
C
1ds, where the curve C is parametrized by r(t) = 〈et,√
2t, e−t〉 for t ∈ [0, 2].
10. Compute the vector line integral
∫
C
F · ds of the vector field F = 〈x2, xy〉 on the line segment from (0, 0) to
(2, 2).
11. Compute the vector line integral
∫
C
F · ds of the vector field F = 〈4, y〉 on the quarter circle x2 + y2 = 1 with
x ≤ 0 and y ≤ 0 oriented counterclockwise.
12. Compute the vector line integral
∫
C
F · ds of the vector field F = 〈x2, xy〉 on the part of the circle x2 + y2 = 9
with x ≤ 0, y ≥ 0 oriented clockwise.
13. Evaluate the integral
∫
C
y dx+ z dy + x dz on the curve parametrized by c(t) = 〈2+ t−1, t3, t2) for o ≤ t ≤ 1.
14. Calculate the line integral of F = 〈ez, ex−y, ey〉over the blue path from P to Q on the figure below.
15. Calculate the work done by the force field F = 〈x, y, z〉 along the path r(t) = 〈cos(t), sin(t), t〉 from 0 ≤ t ≤ 3π.
4
Answers
1. (a) ds = 2√
11dt
(b)26√
11
3
2. (a) ds = 〈1,−t−2〉dt
(b) −1
2
3.√
2(π + π3/3)
4.π4
4
5.14
5
6.1
576(653/2 − 23/2)
7.
√3
2(e− 1)
8.1
2(e2 + 1)
9. e2 − e−2
10.16
3
11. 4.5
12. 0
13.41
10
14. 2− e−1 − e
15.9π2
2
5
3 Lecture 25: The fundamental theorem of vector line integrals
Summary of Lecture
1. A curve parametrized by r(t), t ∈ [a, b] is called closed if r(a) = r(b). (intersects at endpoints). A curve iscalled simple if it intersects itself only at the endpoints. The following figure illustrates a simple closed curve,a simple non-closed curve, a non-simple, non-closed curve and a non-simple closed curve, respectively
2. a region is open if it consists only of interior points (that is, it does not contain its boundary points.)
3. a region (in R2 or R3) is path connected if any two points can be connected by a continuous curve lying in R.
4. A path connected region R (in R2 or R3) is simply connected if every closed curve in R encloses only pointslying in R. (in other words, every closed curve in R can be “shrunk to a point” within R).
R,R2,R3, open/closed balls and R3 \ {0, 0, 0} are all simply connected. R2 \ {0, 0} and annular regions in 2dimensions are not simply connected because they have “2 dimensional holes”.
5. The circulation of a vector field F along a smooth, closed curve C is denoted by the integral
∮
C
F ·T ds
6. Fundamental Theorem of vector line integrals If a vector field F is conservative and φ is the potentialfunction (so that F = ∇φ), then the line integral of F along any path C from a point a to b is given by∫
C
F · dr = φ(b)− φ(a) . In particular, if C is a closed curve, then
∮
C
F · dr = 0.
7. If the domain of a vector field F is simply connected, then F is conservative if and only if F is irrotational.
8. To determine the potential function for a conservative vector field F = 〈F1, F2, F3〉 partially integrate F1 withrespect to x, F2 with respect to y, F3 with respect to z and compare.
6
Exercises
1. Evaluate the line integral of F = 〈3, 6y〉 over the path given by r(t) = 〈t, 2t−1〉 for 1 ≤ t ≤ 4.
2. Evaluate the line integral of F = 〈yez, xez, xyez〉 over the curve c(t) = 〈t2, t3, t− 1〉 for 1 ≤ t ≤ 2.
3. Find a potential function for F = 〈z, 1, x〉.
4. Find a potential function for F = 〈x, y〉 or determine it’s not conservative.
5. Find a potential function for F = y2i + (2xy + ez)j + yezk or determine it is not conservative
6. Find a potential function for F = 〈cos(xz), sin(yz), xy sin(z)〉 or determine it is not conservative.
7. Find a potential function for F = 〈yzexy, xzexy − z, exy − y〉.
8. Evaluate∫C
2xyz dx+ x2z dy + x2y dz over the path c(t) = 〈t2, sin(πt/4), et2−2t〉 for 0 ≤ t ≤ 2.
9. Evaluate∮C
sin(x) dx+ z cos(y) dy + sin(y) dz Where C is the ellipse 4x2 + 9y2 = 36, oriented clockwise.
10. Let F =
⟨1
x,−1
y
⟩. Calculate the work against F required to move an object from (1, 1) to (3, 4) along any
path in the first quadrant.
11. The vector field F =
⟨x
x2 + y2,
y
x2 + y2
⟩is defined on the domain D = {(x, y)|x, y 6= 0}.
(a) Is D simply connected?
(b) Show that F satisfies the cross partial condition. Does this guarantee that F is conservative?
(c) Show that F is conservative by finding a potential function.
(d) Does these results contradict the highlighted statement 7 in Lecture 23 summary?
12. The vector field F =
⟨ −yx2 + y2
,x
x2 + y2
⟩is defined on the domain D = {(x, y)|x, y 6= 0}.
(a) Show that F satisfies the cross partial condition. Does this guarantee that F is conservative?
(b) Is F conservative?
(c) Does these results contradict the highlighted statement 7 in Lecture 23 summary?
7
Answers
1. −9
4
2. 32e− 1
3. V (x, y, z) = xz + y
4. Not Conservative
5. V (x, y, z) = y2z + ezy
6. Not Conservative
7. V (x, y, z) = zexy − yz
8. 16
9. 0
10. ln 4− ln 3
11. (a) no
(b) can easily verify
(c) V (x, y) =1
2ln(x2 + y2)
(d) No, conservative vector fields can be defined in non simply connected regions.
12. (a) Can easily verify
(b) No
(c) No, because the domain is not simply connected.
8
4 Lecture 26: Surface parametrization and scalar surface integrals
Summary of Lecture
1. A surface can be parametrized by using 2 parameters (variables). A general equation of a surface is r(u, v) = 〈f(u, v), g(u, v), h(u, v)〉,where f, g and h are functions of u and v.
2. If a surface is parametrized by r(u, v), the vector ru× rv is a normal vector to the surface at any given point.Also the vector −ru × rv is another normal. An orientation of the surface is a choice of one normal vectorconsistent with the given problem.
3. Some important examples for surfaces are
Surface Parametrization
Sphere (of radius R)r(φ, θ) = 〈R cos(θ) sin(φ), R sin(θ) sin(φ), R cos(φ)〉where θ ∈ [0, 2π] and φ ∈ [0, π]
Cylinder (height H and radius Rr(θ, z) = 〈R cos(θ), R sin(θ), z〉 where θ ∈ [0, 2π],and z ∈ [0, H]
Right circular cone (height H)r(u, v) = 〈u cos(v), u sin(v), u〉 where u ∈ [0, 1] andv ∈ [0, 2π]
Helicoid (height H)r(u, v) = 〈u cos(v), u sin(v), v〉where u ∈ [−1, 1]and v ∈ [0, 2π]
Any graph of explicitly defined function z =k(x, y)
r(x, y) = 〈x, y, k(x, y)〉
From the last example, the parametric equation of a paraboloid is r(x, y) = 〈x, y, x2 + y2〉, and that of aparabolic hyperboloid is r(x, y) = 〈x, y, x2 − y2〉
4. The surface integral of a scalar function f(x, y, z) on a surface S is defined by the limit
x
S
f(x, y, z) ds = lim∆u,∆v→0
n∑
i=1
m∑
j=1
f(r(Pij)‖ru × rv‖∆u∆v
5. The computational formula for evaluating the surface integral of a scalar function f(x, y, z) over a surfacer(u, v) on a region R is given by
x
S
f(x, y, z) ds =x
R
f(r(u, v))‖ru × rv‖ du dv
Note that the right hand side is a double integral over a region in the uv-plane.
6. The surface area of the surface r(u, v) on a region R is given byx
R
1‖ru × rv‖ du dv
7. If the surface is explicitly defined in the form of z = k(x, y), then ‖ru × rv‖ =√
1 + k2x + k2
y and the scalar
surface integral can be computed easily by the formulax
S
f(x, y, z) ds =x
R
f(k(x, y))√
1 + k2x + k2
y dx dy
9
Exercises
1. Let G(x, y) = 〈x, y, xy〉
(a) Calculate Gx, Gy and n(x, y)
(b) Let S be the part of the surface with parameter domain D = {(x, y) : x2 + y2 ≤ 1, x ≥ 0, y ≥ 0}, verifythe following formula and evaluate using polar coordinates.
x
S
1dS =x
D
√1 + x2 + y2 dx dy
(c) Verify the following formula and evaluate
x
S
z ds =
∫ π/2
0
∫ 1
0
(sin(θ) cos(θ))r3√
1 + r2 dr dθ
2. Find the normal vector to the surface G(u, v) = 〈2u+ v, u− 4v, 3u〉 at the point u = 1, v = 4. Also find theequation of the tangent plane to G at that point.
3. Find the normal vector to the surface G(r, θ) = 〈r cos(θ), r sin(θ), 1− r2〉 at the point r = 1/2, θ = π/4. Alsofind the equation of the tangent plane to G at that point.
4. Calculate the scalar surface integralsSf(x, y, z) dS for the function f(x, y, z) = x(x2 + y2) on the surface
G(u, v) = 〈u cos(v), u sin(v), u〉 where 0 ≤ u ≤ 1, 0 ≤ v ≤ 1
5. Calculate the scalar surface integralsSf(x, y, z) dS for the function f(x, y, z) =
√x2 + y2 on the surface
G(r, θ) = 〈r cos(θ), r sin(θ), θ〉 where 0 ≤ r ≤ 1, 0 ≤ θ ≤ 2π
6. Calculate the scalar surface integralsSf(x, y, z) dS for the function f(x, y, z) = 1 on the surface y = 9− z2
where 0 ≤ x ≤ z ≤ 3
7. Calculate the scalar surface integralsSf(x, y, z) dS for the function f(x, y, z) = z on the part of the plane
x+ y + z = 1, where x, y, z ≥ 0
8. Calculate the scalar surface integralsSf(x, y, z) dS for the function f(x, y, z) = z2 on part of the plane
x+ y + z = 0 contained in the cylinder x2 + y2 = 1
9. Calculate the scalar surface integralsSf(x, y, z) dS for the function f(x, y, z) = z on the part of the surface
z = x3 where 0 ≤ x ≤ 1, 0 ≤ y ≤ 1
10. Calculate the scalar surface integralsSf(x, y, z) dS for the function f(x, y, z) = xy+ez on the triangle shown
below.
11. Find the surface area of the portion S if the cone z2 = x2 + y2 where z ≥ 0, contained within the cylindery2 + z2 ≤ 1.
10
Answers
1. (a) Gx = 〈1, 0, y〉, Gy = 〈0, 1, x〉 n(x, y) = 〈−y,−x, 1〉
(b)(2√
2− 1)π
6
(c)
√2 + 1
15
2. 〈12, 3,−9〉 Plane 4x+ y − 3z = 0
3. 〈2r2 cos θ, 2r2 sin θ, r〉 Plane 2√
2x+ 2√
2y + 4z = 5
4.
√2
5
5.2π(2√
2− 1
3
6.37√
37− 1
12
7.
√3
6
8.π√
3
2
9.1
54(10√
10− 1)
10. 3e3 − 6e2 + 3e+ 1
11. π
11
5 Lecture 27: Surface integrals of vector fields
Summary of Lecture
1. The “amount” of the vector field passing through the surface is called the flux of the vector field across thesurface. For a vector field F defined on a surfec S having the normal n, the flux is calculated by summing up(integrating) the component of F parallel to n at each point of the surface S.
2. When choosing the orientation for the surface S having the parametric equation r(u, v) (you have two choicesto start with, ru × rv or −ru × rv), follow these steps;
(a) See whether the orientation is specified in the problem, if so, pick that orientation.
(b) If the surface is a closed surface, pick the orientation so that the normal vector points outward from thesurface.
(c) If the surface is not closed, pick the orientation so that the 3rd component of the normal vector is positive.
In all the formulas given below, a star ∗ indicates “assuming correct orientation”.
3. The flux of a vector field F on a surface S is denoted by the integral
x
S
F · n ds =x
F · dS
4. For a surface S having the parametric equation r(u, v) where u ∈ [a, b] and v ∈ [c, d], the computationalformula for the flux over S is given by
x
S
F · n ds =
∫ d
v=c
∫ b
u=a
F(r(u, v)) · (ru × rv)∗ du dv
5. For an explicitly defined surface z = k(x, y) x ∈ [a, b], y ∈ [c, d], the normal vector is 〈−kx,−ky, 1〉 (for theorientation with z component positive) and the flux can be computed by the easier formula
x
S
F · n ds =
∫ d
y=c
∫ b
x=a
F(x, y, k(x, y) · 〈−kx,−ky, 1〉 dx dy
6. Some applications of the flux are
(a) Outward flux of heat
(b) Flow rate of a fluid across a surface.
12
Exercises
1. Compute the fluxx
S
F · dS for the function F = 〈y, z, x〉 on the plane 3x− 4y + z = 1 with upward pointing
normal , where x, y ∈ [0, 1]
2. Compute the fluxx
S
F · dS for the function F = 〈ez, z, x〉 on the surface parametrized by G(r, s) = 〈rs, r+s, r〉
oriented by Gr ×Gs, where r, s ∈ [0, 1]
3. Compute the fluxx
S
F · dS for the function F = 〈0, 3, x〉 on the part of the sphere of radius 3 on the first
octant with outward pointing normal.
4. Compute the fluxx
S
F · dS for the function F = 〈x, y, z〉 on the part of the unit sphere with outward pointing
normal, where 1/2 ≤ z ≤√
3/2
5. Compute the fluxx
S
F · dS for the function F = 〈z, z, x〉 on the surface z = 9−x2− y2 with upward pointing
normal, where x, y, z ≥ 0
6. Compute the fluxx
S
F · dS for the function F = 〈sin(y), sin(z), yz〉 on the rectangle 0 ≤ y ≤ 2, 0 ≤ z ≤ 3 in
the yz plane, normal pointing in the negative x direction.
7. Compute the fluxx
S
F · dS for the function F = 〈y2, 2,−x〉 on the portion of the plane x + y + z = 1 with
upward pointing normal in the first octant.
8. Compute the fluxx
S
F · dS for the function F = 〈x, y, ez〉 on the cylinder x2 + y2 = 4 with outward pointing
normal , where 1 ≤ z ≤ 5
9. Compute the fluxx
S
F · dS for the function F = 〈xz, yz, z−1〉 on the disk of radius 3 at height 4 parallel to
the xy plane, with upward pointing normal.
10. Compute the fluxx
S
F · dS for the function F = 〈xy, y, 0〉 on the cone z2 = x2 +y2 , where x2 +y2 ≤ 4, z ≥ 0
with downward pointing normal
11. Compute the fluxx
S
F · dS for the function F = 〈0, 0, ey+z〉 on the boundary of the unit cube 0 ≤ x ≤ 1, 0 ≤
y ≤ 1, 0 ≤ z ≤ 1.
12. Compute the fluxx
S
F · dS for the function F = 〈0, 0, z2〉 on the helicoid parametrized by G(u, v) =
〈u cos(v), u sin(v), v〉 , where 0 ≤ u ≤ 1 and 0 ≤ v ≤ 2π with upward pointing normal.
13
Answers
1. −4
2. 4/3− e
3.27
12(3π + 4)
4. π(1−√
3)
5. 312.25
6. 3(cos 2− 1)
7.11
12
8. 32π
9.9π
4
10.8π
3
11. (e− 1)2
12.4π3
3
14
6 Lecture 28: Green’s Theorem
Summary of Lecture
1. The Green’s Theorem is used in two situations,
(a) To evaluate a vector line integral (circulation) enclosing a planar region of the xy plane, by convertingit to a double integral.
(b) To evaluate the area of an irregular region on the xy-plane by using a vector line integral.
2. Green’s Theorem: If ∂D is a smooth closed curve oriented counterclockwise that encloses a connected, simplyconnected region D on the plane, and F = 〈f, g〉 is a vector field where g and h are continuously differentiableon D, then ∮
∂D
F · dr =x
D
(gx − fy) dA
Note: Green’s theorem is valid even if ∂D is a piecewise smooth curve. (eg: a triangle). 3 dimensional version(generalization) of the Green’s Theorem is called Stoke’s Theorem.
3. A 2 dimensional vector field F = 〈f, g〉 is called irrotational if gx − fy = 0
4. If F = 〈f, g〉 is any vector field such that gx − fy = 1, then the area of the region D with a boundary ∂D is
given by A =
∮
∂D
F · dr .
There are many such vector fields F, (eg: F = 〈0, x〉, F = 〈−y, 0〉, F = 〈−y/2, x/2〉)
ExercisesFor the following exercises (1) - (8), use Green’s theorem to evaluate the line integral. Orient the curve counter-clockwise unless otherwise indicated.
1.∮Cy2 dx+ x2 dy where C is the boundary of the unit square x, y ∈ [0, 1]
2.∮Ce2x+y dx+ e−y dy where C is the triangle with vertices (0, 0), (1, 0) and (1, 1).
3.∮Cx2ydx where C is the unit circle centered at the origin.
4.∮CF · dS where F = 〈x+ y, x2− y〉 where C is the boundary of the region enclosed by y = x2 and y =
√x for
0 ≤ x ≤ 1
5.∮CF · dS where F = 〈x2, x2〉 where C consists of the arcs y = x2 and y = x for 0 ≤ x ≤ 1.
6.∮C
(ln(x) + y) dx− x2 dy where C is the rectangle with vertices (1, 1), (3, 1) (1, 4) and (3, 4)
7.∮CF · dS for F = 〈ex+y, ex−y〉 where C is the curve consisting of line segments joining the points (2, 2), (4, 2),
(2, 0) and back to (0, 0), oriented clockwise
8.∮Cxy dx(x
2 + x) dy where C is the triangle with vertices (−1, 0), (1, 0), (0, 1) oriented clockwise.
9. Use Green’s Theorem to calculate the area of a circle of radius 3 centered at the origin.
10. Use Green’s Theorem to calculate the area of the triangle with vertices (0, 0), (1, 0) and (1, 1)
11. Use Green’s Theorem to calculate the area of the region between the x-axis and the cycloid parametrized byc(t) = 〈t− sin(t), 1− cos(t)〉 (cycloid) for 0 ≤ t ≤ 2π
12. Use Green’s Theorem to calculate the area of the region between the graph of y = x2 and the x-axis for0 ≤ x ≤ 2.
15
13. Let x3 + y3 = 3xy be the follum of Descartes
(a) Show that the follum has a parametrization in terms of t = y/x given by x =3t
1 + t3, y =
3t2
1 + t3t ∈ (−∞,∞), t 6== −1
(b) Show that x dy − y dx =9t2
(1 + t3)2dt
(c) Find the area of the loop of the follum
Answers
1. 0
2. e2/2− e3/3− 1/6
3. −π/4
4. −1/30
5. 1/6
6. −30
7.(e2 − 1)(5− e4)
2
8. 1
9. 9π
10. 1/2
11. 3π
12. 8/3
13. (c) : 3/2
16
7 Lecture 29: Stoke’s Theorem
Summary of Lecture
1. The Curl of a vector field F = 〈F1, F2, F3〉 is given by the symbolic cross product:
CurlF = ∇× F =
∣∣∣∣∣∣∣∣∣∣∣
i j k
∂
∂x
∂
∂y
∂
∂z
F1 F2 F3
∣∣∣∣∣∣∣∣∣∣∣
The symbol ∇ =
⟨∂
∂x,∂
∂y,∂
∂z
⟩is called the del operator. A vector field F is called irrotational if ∇×F = 0
2. Conservative vector fields are irrotational, but not all irrotational vector fields are conservative. But anirrotational vector field defined on a simply connected domain is always conservative.
3. The Stoke’s Theorem is used mainly in 2 cases
(a) To evaluate the flux of the curl of a certain vector field by converting it to a vector line integral.
(b) To evaluate a vector line integral (usually a union of lines having different parametrizations, like a tri-angle) as a single vector surface integral.
Also, if we can find a vector potential G to a vector field F (so that F = ∇×G), then we can evaluatethe flux of F across any surface as a line integral using the Stoke’s theorem.
(c) Stoke’s Thoerem Let S be a smooth oriented surface in R3, with a (piecewise) smooth closed boundarycurve ∂S whose orientation is consistent with that of S. Assume that F = 〈F1, F2, F3〉 is a vector fieldswhose componets are continuously differentiable on S, then
∮
∂S
Fdr =x
S
(∇× F) · n ds
In particular, if S is a closed surface, thensS
(∇× F) · n ds = 0
Exercises
1. Find ∇× F for F =⟨z − y2, x+ z3, y + x2
⟩
2. Find ∇× F for F =⟨yx,y
z,z
x
⟩
3. Find ∇× F for F = 〈ey, sin(x), cos(x)〉
4. Find ∇× F for F =
⟨x
x2 + y2,
y
x2 + y2, 0
⟩
5. Compute the flux of curl(F) of the vector field F = 〈ez2 − y, ez2 + x, cos(xz)〉 on the upper hemesphere usingStoke’s Theorem.
17
6. Compute the flux of curl(F) of the vector field F = 〈x + y, z2 − 4, x√y2 + 1〉 over the surface of the wedge
shaped box in the following figure with outward pointing normal.
7. Let I be the flux of F = 〈ey, 2xex2
, z2〉 through the upper hemesphere S of the unit sphere.
(a) Let G = 〈ey, 2xex2
, 0〉. Find a vector field A such that ∇×A = G
(b) Use Stoke’s Theorem to show that the flux of G through S is zero. (Hint: Calculate the circulation of Aaround ∂S)
(c) Calculate I. (Hint: Use (b) to show thatI is equal to the flux of (0, 0, z2) through S.
8. Let F = 〈−x2y, x, 0〉. Referring to the following figure, let C be the closed curve ABCD. Use Stoke’s Theoremto evaluate
∫CF · dS in two ways. First, regard C as the boundary of the rectangle with vertices A,B,C, and
D. Then treat C as the boundary of the wedge shaped box with open top.
9. Let F = 〈−z2, 2zx, 4y − x2〉 and let C be a simple closed curve in the plane x + y + z = 4 that encloses a
region of area 16 (see figure below). Calculate
∮
C
F · ds where C is oriented in the counterclockwise direction
when viewed from above the plane.
10. Let F = 〈0,−z, 1〉. Let S be the spherical cap x2 + y2 + z2 ≤ 1 where z ≥ 1/2. Evaluatex
S
F · dS directly as
a surface integral. Then verify that F = ∇×A, where A = 〈0, x, xz〉 and evaluate the surface integral againusing the Stoke’s Theorem.
18
Answers
1. 〈1− 3z2, 1− 2x, 1 + 2y〉
2. 〈y/z2, z/x2,−1/x〉
3. 〈0, sin(x), cos(x)− ey〉
4. 〈0, 0, 0〉
5. 2π
6. 1/2
7. (a) A = 〈0, 0, ey − ex〉 (b) 0 (c)π/2
8. 234
9. 32
10. 3π/4
19
8 Lecture 30: The Divergence Theorem
Summary of Lecture
1. The divergence of a vector field F = 〈F1, F2, F3〉 is given by the symbolic dot product
div(F) = ∇ · F =∂F1
∂x+∂F2
∂y+∂F3
∂z
This results in a scalar valued function. If for a vector field F, div(F) = 0, we say F is incompressible.
2. For any vector field F with continuous second order partial derivatives, ∇ · (∇× F) = 0.However, if for a given vector field G such that div(G) = 0, G = curl(F) for some vector field F only if G isdefined on a simply connected region.
3. The divergence Theorem is used to evaluate a vector surface integral (flux) over a closed, bounded surface S,by converting it to a triple integral.
4. The divergence Theorem: Let F be a vector field defined on a connected, simply connected region D in R3
enclosed by a smooth oriented surface S. Also assume that the components of F are continuously differentiable.Then x
S
F · n ds =y
D
(∇ · F)dV
Where n is the outward unit normal vector to S.
Exercises
Use the divergence Theorem to evaluate the following surface integrals for the function F given on the surfaceS .
1. F = 〈0, 0, z3/3〉 where S is the sphere x2 + y2 + z2 = 1
2. F = 〈y, z, x〉 where S is the sphere x2 + y2 + z2 = 1
3. F = 〈x3, 0, z3〉 S is the first octant of the sphere x2 + y2 + z2 = 4
4. F = 〈ex+y, ex+z, ex+y〉 where S is the boundary of the unit cube x, y, z ∈ [0, 1]
5. F = 〈x, y2, z + y〉 where S is the boundary of the region contained in the cylinder x2 + y2 = 4 between theplanes z = x and z = 8
6. F = 〈x2 − z2, ez2 − cos(x), y3〉 where S is the boundary of the region bounded by x + 2y + 4z = 12 and the
coordinate planes in the first octant.
7. F = 〈x+ y, z, z − x〉 where S is the boundary of the region between the paraboloid z = 9− x2 − y2 and thexy-plane.
8. F = 〈ez2 , 2y + sin(x2z), 4z +√x2 + 9y2〉 where S is the region x2 + y2 ≤ z ≤ 8− x2 − y2
9. Calculate the flux of the vector field F = 2xyi− y2j + k through the surface S in the following figure.
20
Answers
1.4π
15
2. 0
3.128π
5
4. (e− 1)2
5. 64π
6. 216
7. 81π
8. 96π
9. π
21
MAC2313 Final A
(5 pts) 1. Let f(x, y, z) be a function continuous in R3 and let S be a surface parameterizedby r(u, v) with the domain of the parameterization given by R; how many of the following are true?(Assume that the order of the product du dv is consistent with the limits in the double integral.)
a. The vector ru × rv lies in the tangent plane of S at a given point.
b.∫ ∫
S f(x, y, z) dS =∫ ∫
R f(r(u, v))|ru × rv| du dv.
c. The vectors ru and rv are orthogonal to the tangent plane of S at a given point.
d. The surface area of S is given by the integral∫ ∫
R |ru × rv| du dv.
A. 0 B. 1 C. 2 D. 3 E. 4
(5 pts) 2. A surface S is given by the graph of z = 3x2 + 3y2 − 5, {(x, y) | x2 + y2 ≤ 9} andoriented so that the z component of the unit normal vector is negative; a unit normal vector to thesurface is given by:
A. (36x2 + 36y2 + 1)−1/2 ⟨6x, 6y, −1⟩ B. (36x2 + 36y2 + 1)−1/2 ⟨−6x, −6y, −1⟩
C. (36x2 + 36y2 + 1)−1/2 ⟨−6x, −6y, 1⟩ D. (36x2 + 36y2 + 1)−1/2 ⟨6x, 6y, 1⟩
E. none of the above
(5 pts) 3. Let S be the part of the surface x + y + z = 1 which lies above the triangular regionof the x, y - plane with vertices (0, 0), (1, 0), and (0, 1) and oriented so that the normal vector hasa positive z component. If F = ⟨2, −3, 1⟩ , the flux of F through S is given by:
A. 0 B. 1/2 C. −1/2 D. −2 E. 2
(5 pts) 4. Let C be the square with vertices (0, 0), (1, 0), (1, 1) and (0, 1) and having a counter-clockwise orientation and let F = ⟨y + x2, ey − x⟩. Which of the following could be used to computethe circulation of F around C?
A. Bubba’s Theorem
B. Clairaut’s Theorem
C. Green’s Theorem
D. the Divergence Theorem
E. the Fundamental Theorem of Line Integrals
(5 pts) 5. The circulation of F around C in problem 4 has the value:
A. 0 B. −2 C. −4 D. −6 E. −8
(5 pts) 6. If F = ⟨x2e3y, e3y cos(πz), z ln x⟩, then the divergence of F at the point (1, 0, 1) is equalto:
A. 0 B. 1 C. 5 D. e3 E. none of the above
(5 pts) 7. If F = ⟨2xy3, xy + zy2, z3(x − 2y)⟩, then the curl of F at the point (1, 2,−1) is equalto:
A. ⟨−2, 1,−22⟩ B. ⟨2, 1,−22⟩ C. ⟨−2, 1,−22⟩ D. ⟨2, −1, 22⟩ E. ⟨−2, −1, 22⟩
(5 pts) 8. If a curve C is given parametrically by r(t) = ⟨3, 2t, t2⟩ , t : 0 →√
3 andf(x, y, z) = 6xy, then the value of
∫C f(x, y, z) ds is:
A. 144 B. 146 C. 168 D. 172 E. none of the above
(5 pts) 9. Let F be a conservative vector field in R3 and let C1 and C2 be two simple, smoothcurves which share the same initial point P and terminal point Q; then
∫C1
F · dr =∫C2
F · dr.
A. true B. false
(5 pts) 10. Let F = ⟨x − y, x + y⟩; the circulation of F around the unit circle with counterclock-wise orientation is equal to:
A. 0 B. 2π C. −2π D. 2π + 1 E. −2π − 1
(5 pts) 11. Let f(x, y, z) be a function continuous in R3 and let S be a surface given as thegraph of the differentiable function z = k(x, y) with domain R; how many of the following are true?(Assume that the order of the product dy dx is consistent with the limits in the double integral.)
a.∫ ∫
S f(x, y, z) dS =∫ ∫
R f(x, y, k(x, y))√
k2x + k2
y + 1 dy dx.
b.∫ ∫
S [f(x, y, z)]2 dS =∫ ∫
R [f(x, y, k(x, y))]2√
k2x + k2
y + 1 dy dx.
c. If r(x, y) = ⟨x, y, k(x, y)⟩, then rx × ry is normal to the surface.
d. The vector ⟨kx, ky, 1⟩ is orthogonal to the tangent plane of S at a given point.
A. 0 B. 1 C. 2 D. 3 E. 4
(5 pts) 12. Let S be the unit sphere centered at the origin and let F = ⟨−2x, 3y, 5z⟩, then theflux across the surface is given by:
A. 0 B. 6π C. 8π D. 12π E. 16π
(5 pts) 13. Let S be the part of the cylinder of radius 5 centered on the z axis with x ≤ 0 and|z| ≤ 1; then S is given parametrically by r(θ, z) = ⟨5 cos θ, 5 sin θ, z⟩ with domainR = { (θ, z) | π ≤ θ ≤ 2π, −1 ≤ z ≤ 1 }.
A. true B. false
(5 pts) 14. If curl F = 0, then F is called irrotational.
A. true B. false
(5 pts) 15. Let F be any continuous vector field given by F = ∇ϕ(x, y); let C be any simple,smooth, open, oriented curve in R2, then
∫
CF · dr = 0.
A. true B. false
(5 pts) 16. If F = ⟨y, x + z2, 2zy⟩ and C is given parametrically by r(t) =⟨2, cos3 t, sin3 t
⟩,
t : 0 → π/2, then what is the value of∫C F · T ds?
A. −3 B. −2 C. −1 D. 0 E. 4
(5 pts) 17. If f(x, y, z) = z(x2 + y2) and S the part of the cone z = 2√
x2 + y2, 0 ≤ z ≤ 4, then∫ ∫S f(x, y, z) dS is equal to which of the following:
A. 2√
5∫ 2π0
∫ 20 r4 dr dθ B. 2
√5
∫ 2π0
∫ 20 r3 dr dθ C. 2
√5
∫ 2π0
∫ 20 r5 dr dθ
D. 5√
2∫ 2π0
∫ 40 r4 dr dθ E. none of the above
(5 pts) 18. If F(x, y, z) = ⟨x2, 3y + x, 2y⟩ and S is a surface given by r(u, v) = ⟨3u, 2u sin v, 0⟩ ,0 ≤ u ≤ 1, 0 ≤ v ≤ π/2 and oriented so that the z component of the normal vector is positive;then
∫ ∫S F(x, y, z) · n̂ dS is equal to:
A. 0 B. 2 C. 4 D. 6 E. 8
(5 pts) 19. Let F be a constant vector field and let S be a smooth, closed surface; then thestatement
∫ ∫S F · n̂ dS = 0 is true:
A. sometimes B. always C. never
(5 pts) 20. Assuming the limits for the integrals are consistent with the given properties, whichof the following are true:
I. Stoke’s Theorem is given by∫ ∫
S (∇ × F) · n̂ dS =∫C F · T ds.
II. If F is constant and S is a smooth, bounded, closed surface, then∫ ∫
S (∇ × F) · n̂ dS = 0.
III. If F is constant and C is a smooth, simple closed curve, then∫C F · dr = 0.
IV. The Divergence Theorem is given by∫ ∫
S ∇ · F dS =∫ ∫ ∫
D F · n̂ dV.
A. only I B. only III C. only I and II D. only I, II, and IV E. I, II, and III
Bonus (5 pts) 21. Let S be the part of the unit sphere with z ≥ 0 oriented so that the z com-ponent of the unit normal vector is positive; if F = ⟨−y/2, x/2, z2(x + y)⟩, then
∫ ∫S (∇ × F) · n̂ dS
is equal to: (Don’t compute this integral directly.)
A. 0 B. −π C. π D. −2π E. 2π
Bonus (5 pts) 22. Let F be a radial vector field given byr
|r|5 ; a potential function for F is:
A. (1/3)(x2 + y2 + z2)−3/2 B. (2/3)(x2 + y2 + z2)−3/2 C. (1/7)(x2 + y2 + z2)−7/2
D. (2/7)(x2 + y2 + z2)−7/2 E. none of the above
MAC2313 Final A
(5 pts) 1. How many of the following are necessarily true?
i. The vector field F = ⟨−2x + 3y, 3x − 5y⟩ is conservative.
ii. The vector field F = 5(x2 + y2)−3/2 ⟨x, y⟩ is radial.
iii. All constant vector fields in R3 are conservative.
iv. Gravitational fields in R3 are conservative and radial.
A. 0 B. 1 C. 2 D. 3 E. 4
(5 pts) 2. How many of the following are necessarily true?
i. The potential function for a conservative vector field is unique.
ii. The divergence of a constant vector field in R3 is equal to the zero vector.
iii. The divergence of a conservative vector field in R3 is a constant.
iv. The curl of a conservative vector field in R3 is equal to the zero vector.
A. 0 B. 1 C. 2 D. 3 E. 4
(5 pts) 3. Let C be the curve given parametrically by r(t) = ⟨t + 3, 4 − 2t⟩ , t : 1 → 2; iff(x, y) = 2x + 4y, the value of
∫C f(x, y) ds is:
A. 0 B. 13√
5 C. 4√
5 D. 26√
5 E. 22√
5
(5 pts) 4. Let C be the curve given parametrically by r(t) = ⟨3t, 1 − 2t2, 4⟩ , t : 0 → 2; thevalue of
∫C y dx + x2 dy + z dz is:
A. 0 B. −40 C. −86 D. −154 E. −210
(5 pts) 5. Let C be the upper half of the unit circle oriented clockwise and let F = ⟨2y + 1,−2x − 1⟩,the value of
∫C F · dr is:
A. 0 B. −2 − π C. 2 + 2π D. −2 − 2π E. 2 + π
(5 pts) 6. Let D = { (x, y) | |x| < 1, |y| < 2}; how many of the following are true?
i. D is simply connected.
ii. D is connected.
iii. D is open.
iv. The boundary curve(s) for D is closed but not simple.
A. 0 B. 1 C. 2 D. 3 E. 4
(5 pts) 7. Let C be a simple, smooth curve with initial point (1, 0, 1) and terminal point(0, 0,−1); if F = ⟨6yz2e2xy, 6xz2e2xy, 6ze2xy + 1⟩, the value of
∫C F · dr is:
A. 0 B. −2 C. −4 D. 3 E. none of the above
(5 pts) 8. How many of the following are necessarily true?
i. The line integral of a conservative vector field is independent of the path connecting an initialpoint P to a terminal point Q.
ii. The curl of a constant vector field in R3 is equal to the scalar zero.
iii. A vector field F = ⟨f(x, y), g(x, y)⟩ is conservative if fx = gy.
iv. The circulation of a conservative vector field along a smooth, oriented curve is equal to zero.
A. 0 B. 1 C. 2 D. 3 E. 4
(5 pts) 9. The part of the unit sphere with x ≤ 0 and z ≤ 0 is parameterized byr(θ, ϕ) = ⟨cosθ sin ϕ, sin θ sin ϕ, cos ϕ⟩; the domain for this parameterization is equal to:
A. D = { (θ, ϕ) | 0 ≤ θ ≤ 2π, 0 ≤ ϕ ≤ π } B. D = { (θ, ϕ) | π/2 ≤ θ ≤ π, 0 ≤ ϕ ≤ π/2 }
C. D = { (θ, ϕ) | π ≤ θ ≤ 2π, 0 ≤ ϕ ≤ π/2 } D. D = { (θ, ϕ) | π/2 ≤ θ ≤ 3π/2, π/2 ≤ ϕ ≤ π }
E. D = { (θ, ϕ) | π ≤ θ ≤ 2π, π/2 ≤ ϕ ≤ π }
(5 pts) 10. Let f(x, y, z) = z + y2 and let S be the surface parameterized byr(u, v) = ⟨2u,−3v, u + v⟩ with 0 ≤ u ≤ 1, 0 ≤ v ≤ 2. The integral
∫ ∫S f(x, y, z) dS is equal to:
A. 0 B. 160 C. 189 D. 84 E. 62
(5 pts) 11. Let S be the surface parameterized by r(u, v) = ⟨2v, u2 + v2, u⟩ with0 ≤ u ≤ 2, 0 ≤ v ≤ 2. A vector normal to the tangent plane of the surface at u = 1, v = 1 is:
A. ⟨−2, 2,−4⟩ B. ⟨1,−2, 2⟩ C. ⟨1, 2, −4⟩ D. ⟨−2, 1,−4⟩ E. ⟨2, 2,−4⟩
(5 pts) 12. Let f(x, y, z) = xyez and let S be the part of the paraboloid z = x2 + y2 + 8 wherex2 + y2 ≤ 5. The integral
∫ ∫S f(x, y, z) dS is equal to:
A.∫ 5−5
∫ √5−x2
−√
5−x2 xyez dy dx B.∫ √
5−
√5
∫ √5−x2
−√
5−x2 xyex2+y2+8 dy dx
C.∫ 5−5
∫ √5−x2
−√
5−x2 xyex2+y2+8√
4x2 + 4y2 + 1 dy dx D.∫ √
5−
√5
∫ √5−x2
−√
5−x2 xyex2+y2+8√
2x + 2y + 1 dy dx
E. none of the above
(5 pts) 13. Let F = ⟨z, x, y⟩ and let S be the surface parameterized by r(u, v) = ⟨cos u, sin u, 2v + 3⟩with domain R = { (u, v) | 0 ≤ u ≤ 2π, 0 ≤ v ≤ 2 } and oriented so that normal vectors to thesurface are pointing away from the z-axis; the integral
∫ ∫S F · n̂ dS is given by:
A.∫ 2π0
∫ 20 6v cos u + 6 sin u dv du B.
∫ 2π0
∫ 20 6 sin u + 6v + v sin u cos u dv du
C.∫ 2π0
∫ 20 4v cos u + 6 cos u + 2 sin u cos u dv du D.
∫ 2π0
∫ 20 6 cos u + sin u cos u dv du
E.∫ 2π0
∫ 20 12v cos u sin u dv du
(5 pts) 14. Let F = ⟨x, y, z⟩ and let S be the part of the plane x + y + z = 1 defined above thetriangular region in the x, y-plane with vertices (0, 0), (1, 0), and (0, 1) and oriented so that the zcomponent of the normal vector is positive; the flux of F across S is given by:
A.∫ 10
∫ 10 1 dy dx B.
∫ 10
∫ −x+10 1 dy dx C.
∫ 10
∫ −x+10 x + y dy dx
D.∫ 10
∫ −x+10 1 − x − y dy dx E. none of the above
(5 pts) 15. Let C be the closed curve in the x, y-plane given by the triangle with vertices (−1, 0),(0, 2), and (1, 0) and oriented counter-clockwise. If F = ⟨4y − 1, 3 − 2x⟩ , then the circulation of Faround C is equal to:
A. 0 B. −6 C. 12 D. −12 E. 6
(5 pts) 16. Let F = ⟨2xy + 4z2, x2 + 2z, 2y + 8xz⟩; the divergence of F at the point(x, y, z) = (1, 1, 1) is equal to:
A. 0 B. 4 C. 10 D. 12 E. 6
(5 pts) Bonus 17. The vector field in problem 16 is conservative.
A. True B. False
(5 pts) 18. Let F = ⟨xz, xy, yz⟩; the curl of F at the point (x, y, z) = (0, 0, 0) is equal to:
A. ⟨−1, −1,−1⟩ B. ⟨1, 1, 1⟩ C. ⟨1,−1, 1⟩ D. ⟨0, 0, 0⟩ E. ⟨1, 1, 1⟩
(5 pts) Bonus 19. The vector field in problem 18 is conservative.
A. True B. False
(5 pts) 20. Let S be a closed surface bounded by a cylinder of radius one centered on the z-axis,z = 0, and z = 2 and let F = ⟨−6y, 2x, z2⟩, then the flux across the surface is equal to:
A. 0 B. π C. 4π D. 3π E. 2π
(5 pts) 21. Let S be the part of the paraboloid z = 9 − x2 − y2 with z ≥ 0 and oriented so
that the z component of the normal vector is positive; if F =⟨−y + z, x, ex2+y2
⟩, then the integral∫ ∫
S (∇ × F) · n̂ dS is equal to:
A. 0 B. 6π C. 3π D. 2π E. 18π
(5 pts) 22. Which of the following are true:
I. The Divergence Theorem is given by∫ ∫ ∫
D F · n̂ dV =∫ ∫
S ∇ · F dS.
II. Stoke’s Theorem is given by∫C F · T ds =
∫ ∫S (∇ × F) · n̂ dS.
III. If F is conservative and S is a smooth bounded surface, then∫ ∫
S (∇ × F) · n̂ dS = 0.
IV. The flux of a constant vector field out of a solid is always greater than zero. (Hint: considerthe Divergence Theorem)
A. only II and III B. only I and III C. only I, II, and IV D. only I, II, and III
E. I, II, III, and IV
MAC2313 Final A
(5 pts) 1. Let C be the bottom half of the circle of radius 3 centered at the origin in the x, y-plane and with a counterclockwise orientation; the integral
∫C (2x2y − y)2 ds is equal to:
A. 9∫ 2ππ (6 cos2 t − sin t)2 dt B. 3
∫ 2ππ (18 cos2 t − sin t)2 dt
C. 3∫ 2ππ (18 cos2 t + 3 sin t)2 dt D.
∫ 2ππ (54 cos2 t + 9 sin t)2 dt
E. none of the above
(5 pts) 2. The function ϕ(x) =k√
x2+y2+z2is a potential function for which of the following
vector fields:
A. F =−2kr|r|3 B. F =
−kr2|r|3 C. F =
−kr|r| D. F =
−kr|r|3 E. F =
−2kr|r|
(5 pts) 3. If F = ⟨2y,−4x, y⟩ and C is given parametrically by r(t) = ⟨et, 3, −t⟩ , t : 0 → 1,then
∫C F · dr is equal to:
A. 2e − 3 B. 6e − 9 C. 6e − 3 D. 2e + 3 E. 6e + 9
(5 pts) 4. Let F ⟨f, g, h⟩ and G = ⟨l, m, n⟩ be a differentiable vector fields and let C be a simple,smooth curve given parametrically by r(t) = ⟨x(t), y(t), z(t)⟩ , t : a → b; which of the following aretrue:
I.∫C F · dr =
∫C f dx + g dy + h dz
II.∫C F · T ds = − ∫
−C F · T ds
III.∫ ba [(f + l)x′(t) + (g + m)y′(t) + (h + n)z′(t)] dt =
∫C F · dr +
∫C G · dr
IV. If F is continuous, then∫C F · T ds = 0
A. only III B. only I and II C. only III and IV D. only I, II, and III
E. I, II, III, and IV
(5 pts) 5. Let D be the annular region in the x, y-plane which lies between the circles x2 +y2 = 1and x2 + y2 = 4; which of the following are true:
I. D is connected.
II. D is simply connected.
III. The curve x2 + y2 = 4 is closed but not simple.
IV. The curve x2 + y2 = 1 is closed and simple.
A. only III B. only IV C. only III and IV D. only I and III E. only I and IV
(5 pts) 6. If a curve C is given parametrically by r(t) = ⟨t2, 3t2⟩ , t : 0 → 1 then the integral∫C y dx + x dy is equal to:
A. −1 B. 0 C. 1 D. 2 E. 3
(5 pts) 7. Let F = ∇ϕ and let C be a smooth curve parametrized by r(t), t : a → b, then∫C F · T ds = ϕ(r(b)) − ϕ(r(a)).
A. true B. false
(5 pts) 8. The vector field F = ⟨2yex + 2xz2, ex + z2 cos y, 2z + sin y + 2zx⟩ is conservative.
A. true B. false
(5 pts) 9. Let F = ⟨18xy2, 18x2y + 12y2⟩ and let C be the square in the x, y-plane with vertices(0, 0), (3, 0), (0, 3), and (3, 3) and with a counter-clockwise orientation; then the circulation of Fabout C is given by:
A. −2/3 B. −1/3 C. 0 D. 1/3 E. 4/9
(5 pts) 10. Let F = ⟨2x, 6y⟩ and let C be a simple, smooth curve with initial point (1, 3) andterminal point (2, 5);
∫C F · T ds is equal to:
A. 23 B. 47 C. 51 D. 61 E. 63
(5 pts) 11. Let a surface S be given parametrically by r(u, v) = ⟨2u + v, 3u, 4v⟩ ,−∞ < u, v < ∞; a normal vector to the surface at (3, 3, 4) is given by:
A. ⟨12,−8, 3⟩ B. ⟨6, −4, 3⟩ C. ⟨−12, 8, 3⟩ D. ⟨−6, 4,−3⟩ E. ⟨−6,−4, 3⟩
(5 pts) 12. Let S be the part of the plane 3x + 3y + z = 6 which lies in the first octant andoriented so that the z component of the normal vector is positive; if F = k, what the value of∫S F · n dS?
A. −1 B. 0 C. 1 D. 2 E. 4
(5 pts) 13. Let a surface S be given parametrically by r(u, v) = ⟨3 cos u, 3 sin u, v⟩ ,0 ≤ u ≤ 2π, 0 ≤ v ≤ 1; if f(x, y, z) = z, then
∫S f(x, y, z) dS is equal to:
A. 0 B. π C. 2π D. 3π E. 4π
(5 pts) 14. The standard orientation of a closed surface has the normal vectors pointing intothe region bounded by the surface.
A. true B. false
(5 pts) 15. Let D be a connected and simply connected bounded region in the x, y-plane andlet ∂D be smooth with a counter-clockwise orientation. If F = ⟨f, g⟩ with f and g differentiable,which of the following are true:
I. If∮∂D F · dr =
∫ ∫D gy − fx dA
II. If gy − fx = 1, then Area(D) =∮∂D F · dr
III. If F is conservative, Area(D) = 0
IV.∮∂D F · dr =
∮∂D f dx + g dy
A. only IV B. only I and II C. only I and III D. only II and III
E. only I, II, and III
(5 pts) 16. If F = ⟨x, y, 1⟩ and S is a surface given by r(u, v) = ⟨u cos v, u sin v, v⟩ , 0 ≤ u ≤ 1,0 ≤ v ≤ 2π oriented so that the z component of the normal vector is positive; then
∫S F · n dS is
equal to which of the following:
A. 0 B. −2π C. 2π D. π E. −π
(5 pts) 17. Let S be a surface given by the graph of z = xey, 0 ≤ x ≤ 1, 0 ≤ y ≤ 1, orientedso that the z component of the normal vector is positive; if F = ⟨xy, 4x2, xyey⟩ then the flux of Facross S is equal to:
A. −1 − e B. −1 + e C. 1 − e D. e + 1 E. 0
(5 pts) 18. Let C be the curve defined as the triangle with counterclockwise orientation withvertices (−2, 0), (2, 0), and (0, 2); if F = ⟨x2 + 2y, x − y2⟩, then
∮C F · dr is equal to: (use Green’s
Theorem)
A. 0 B. −4 C. −6 D. −8 E. −10
(5 pts) 19. Let S be a smooth oriented surface which bounds the solid R and let F be a vectorfield whose components have continuous first partial derivatives in R; the Divergence Theorem isgiven by:
A.∫ ∫
S F dS =∫ ∫ ∫
R div F dV
B.∫ ∫
S div F dS =∫ ∫ ∫
R F · n̂ dV
C.∫ ∫
S F · n̂ dS =∫ ∫ ∫
R div F dV
D.∫ ∫
S ∇ · F dS =∫ ∫
S F · n̂ dS
E.∫ ∫
S F · n̂ dS =∫ ∫ ∫
R (div F) · n̂ dV
(5 pts) 20. Let D be the box in the first octant bounded by the three coordinate planes and theplanes z = 1, x = 2, and y = 3 and let F = ⟨x, z, y⟩; calculate the flux out of the surface of the box(Hint: use the divergence theorem.)
A. 6 B. 1 C. 0 D. 12 E. 3
Bonus (5 pts) 21. Let S be a hemisphere given parametrically by r(θ, ϕ) = ⟨cos θ sin ϕ, sin θ sin ϕ, cos ϕ⟩with a domain given by R = {(θ, ϕ) | 0 ≤ θ ≤ 2π, 0 ≤ ϕ ≤ π/2} and suppose the surface is orientedso that the z component of the normal vector is positive; if F = ⟨−x, y, 3⟩, what is value of theintegral
∫ ∫S (∇ × F) · n̂ dS?
A. 0 B. 1 C. 2π D. −1 E. −2π
Bonus (5 pts) 22. Let F be a vector field with differentiable component functions; if curl F = 0on a simply connected domain, then the vector field is conservative on that domain.
A. true B. false
I made the following practice exams about an year ago for cy summer calc III class. I do not have the workedout solutions for them. The answers are in the last page.
67
Fall 2014FINAL: REVIEW 1
1. Evaluate the line integral
∫
C(2xy + 5)dx+ (x2 − 4z)dy − (4y)dz where C is parametrized
by c(t) = (t sin πt, (t2 + 1) cosπt, t2 + t) for 0 ≤ t ≤ 1
(a) −8
(b) 16
(c) 8π
(d) −16
(e) 0
2. If the curve C is given by c(t) = (cos t, sin t, t) where 0 ≤ t ≤ π and f(x, y, z) = xy+ z,
the value of
∫
Cf(x, y, z)ds is given by;
(a) π/2
(b) −π2
(c) π2/√
2
(d) π2/2
(e)√
2π2
3. Compute∫F. dS for the vector function F = 〈4, y〉 for the quarter circle x2 + y2 = 1
with x ≤ 0, y ≤ 0 oriented counterclockwise.
(a) −4
(b) −1
(c) 9
(d) 4.5
(e) 0
4. Let F = 〈3zy−1, 4x,−y〉 and let C be the curve given parametrically by r(t) = (et, et, t)
for t ∈ [−1, 1]. Then
∫
CF . T ds is equal to;
(a) 0
(b) e4 − 1
(c) e2 − e−2 − e+ e−1
(d) e2 − e−2 − 2e+ 2e−1
(e) 2e2 − 2e−2 − e+ e−1
5. Find a potential function V for F or determine that F is not conservative.
F = 〈yzexy, xzexy − z, exy − y〉
(a) F is not conservative
(b) V (x, y, z) = zey − yz + C
(c) V (x, y, z) = zexy − yz + C
(d) V (x, y, z) = zex + yz + C
(e) V (x, y, z) = zexy + yz + C
6. Let F be a two dimensional vector field and C,D be two simple smooth curves in R2
going from P to Q. Which of the following statements are true.
A. If F is conservative, then
∫
CF. dS =
∫
DF. dS.
B. If P = Q, then
∫
CF. dS = 0
C. The integral
∫
CF. dS gives the net change in potential.
D. If F is conservative and C is closed, the circulation of F around C is zero.
(a) A only
(b) A and D only
(c) A, C and D only
(d) A, B and C only
(e) B and C only
7. If a curve C is given parametrically by r(t) = 〈cos t, sin t〉, where 0 ≤ t ≤ 2π andF = 〈x, 0〉, then the flux of F across C is given by;
(a) 0
(b) π
(c) −π(d) 2π
(e) −2π
2
8. Let F be a two dimensional vector field and let C be a simple smooth closed curve inR2; which of the following statements is/are true?
i. If C is oriented clockwise, then the normal vector textbfn points outwardfrom the curve.
ii. The circulation of F around C is given by
∫
CF.nds
iii. The flux of F around C is given by
∫
CF.Tds
iv. If F is conservative, then
∫
CF.Tds = 0
(a) only iv
(b) only i and iv
(c) only iii and iv
(d) only i, ii and iv
(e) all four statements are true.
9. Let F =
⟨ −yx2 + y2
,x
x2 + y2
⟩in R2 \ {0}. Find the circulation of F arounf the unit
circle oriented counterclockwise.
(a) 0
(b) 1
(c) π
(d) −π(e) −1
10. G(r, θ) = (r cos θ, r sin θ, 1− r2) parametrizes the paraboloid z = 1− x2 − y2.
(a) TRUE
(b) FALSE
11. Let S be the portion of the sphere x2 + y2 + z2 = 9 where 1 ≤ x2 + y2 ≤ 4 and z ≥ 0.What is the area of S? (Hint: Find a parametrization of S in polar coordinates)
(a) π ln(0.8)
(b) π ln(1.2)
(c) 6π ln(1.6)
(d) 3π ln(1.6)
(e) 2π ln(1.2)
3
12. Let f(x, y, z) =x2
4z. Find the surface integral
∫∫
Sf(x, y, z) dS for the surface given
by z = 4− x2 − y2 where 0 ≤ z ≤ 3.
(a) π/6(17√
17− 5√
5)
(b) π/12(17√
17− 5√
5)
(c) π/2(17√
17− 5√
5)
(d) −π/12(17√
17− 5√
5)
(e) π/12(5√
5))
13. Let G(u, v) = (u, v3, u+ v), where 0 ≤ u ≤ 1, 0 ≤ v ≤ 1. Find
∫∫
Sy dS
(a)19√
19− 1
108
(b)6√
6− 1
54
(c)19√
19 + 1
108
(d)19√
19− 1
72
(e)
√19− 1
108
14. Calculate
∫
Cxy dx + (x2 + x) dy where C is the triangular path traversed in the order
of the vertices (0, 1), (−1, 0), (1, 0).
(a) 3
(b) 2
(c) −1
(d) 0
(e) 1
15. Calculate the area of the region between the x axis and the cycloid parametrized byc(t) = 〈t− sin t, 1− cos t〉 for 0 ≤ t ≤ 2π
(a) 2π
(b) 3π
(c) 6π
(d) π
(e) 4π
4
16. Find the surfac integral
∫∫
SF.dS for F = 〈ez, z, x〉 where S is the surface parametrized
by G(r, s) = (rs, r + s, r) oriented by Tr ×Ts.
(a) 4/3− e(b) e− 1/3
(c) 4/3− 1/e
(d) e2
(e) 0
17. Calculate∫∫S zdS where S is the part of the surface z = x3 where x, y ∈ [0, 1].
(a)5√
10
27+
1
54
(b)5√
10
27
(c)1
54(d) 0
(e)5√
10
27− 1
54
18. Find
∫∫
SF. dS for the vector field F = 〈−y, z,−x〉 on the surface x2 + y2 + z4 = 4,
z ≥ 0 with outward pointing normal.
(a) 27
(b) 54
(c) -27
(d) 9
(e) -3
19. Use the Stokes theorem to comupute the flux of curl(F) through the upper hemespherex2 + y2 + z2 = 1, z ≥ 0 with outward pointing normal for F = 〈ez2 − y, ez3 +x, cos(xz)〉
(a) 0
(b) −π(c) eπ
(d) 2π
(e) π
5
20. Let F = 〈x, y2, z+y〉 and let S be the boundary of the region contained in the cylinderx2 + y2 = 4 between the planes z = x and z = 8. Find the flux of F across the surfaceof S
(a) 64π
(b) 0
(c) 32π
(d) −64π
(e) 11π
21. Let F = 〈z sec2 x, z, y + tan x〉, and let S be the surface of the upper ellipsoid 9x2 +y2 + 4z2 = 36 , z ≥ 0. find the circulation of F around the boundary of S
(a) 0
(b) π
(c) pi
(d) 8π + 2√
2
(e) 8π − 2√
2
22. Let F = 〈z,−z,−y2〉 and let C consist of the three line segments that bound the planez = 8 − 4x − 2y in the first octant, and suppose that C is oriented counterclockwisewhen viewed from above. The value of
∫C F. dr is given by:
(a) 0
(b) −88
3
(c) −71
6
(d) −23
3
(e) −71
3
6
Answer Key
Multiple Choice Questions
1. B2. C3. D4. E5. C6. B7. B8. D9. C10. A11. D12. B13. A14. E15. C16. A17. E18. B19. D20. A21. A22. B
7
Fall 2014FINAL: REVIEW 2
1. Let C be the left half of the circle of radius 2 centered at (3, 2) with counterclockwise
orientation. The integral
∫
Cx2 − 6x+ y2 − 4y + 13 ds is equal to
(a) 4π
(b) 8π
(c) −2π
(d) 2π
(e) π
2. Calculate
∫
CF . Tds for the vector field F = 〈ez, ex−y, ey〉 on the curve given by the
path from P to Q in the figure below.
(a) 2− e2
(b) 2− e2 − e−1
(c) 2− e− e−1
(d) e− e−1
(e) 2− e−1
3. Find a potential function for the vector field below or show that it is not conservative.
F =
⟨y
1 + x2, tan−1 x, 2z
⟩
(a) V (x, y, z) = y tan−1(x)− z2 + C
(b) V (x, y, z) = y2 tan−1(x)− z2 + C
(c) V (x, y, z) = y tan−1(x) + z + C
(d) V (x, y, z) = y tan−1(x) + z2 + C
(e) The vector field F is not conservative
4. Calculate Curl(F) for F = 〈x2 + y2, ln√y2 + 1 + z2, z3 sin(z2)ez
3〉.
(a) 〈2z,−2y, 0〉(b) 〈0, 0, 0〉(c) 〈2z, 0,−2y〉(d) 〈2z, 2y, 0〉(e) 〈2z, 0, 2y〉
5. Let f(x, y, z) = 6xz− 2y2. Calculate
∫
Cf(x, y, z) ds along the curve (t, t2/
√2, t3/3) as
t goes from 0 to 2
(a) 0
(b) π
(c) 864/35
(d) 123/63
(e) 8π
6. For the vector field v = 〈ey, 2x− 1〉, calculate the flux across the parabola y = x2 for0 ≤ x ≤ 1 oriented from left to right.
(a) 0
(b) e− 1
(c) 1
(d) 1− e(e) e2
7. Calculate the work done by the force field F = 〈x, y, z〉 along the path c(t) = (cos t, sin t, t)for 0 ≤ t ≤ 6π.
(a) 9π2
(b) 18π2
(c) 9π
(d) 36− 9π2
(e) 0
2
8. If a curve C is given parametrically by r(t) = 〈cos t, sin t〉, t ∈ [0, 2π] and F = 〈x, 0〉,then the flux across C is given by
(a) 0
(b) −π(c) 2π
(d) π
(e) −2π
9. Calculate the line integral
∫
Cy dx− x dy on the parabola y = x2 for 0 ≤ x ≤ 2.
(a) 1
(b) 0
(c) −8/3
(d) −4/3
(e) 6
10. Evaluate
∫
C2xyz dx+ x2z dy + x2y dz over the path c(t) = (t2, sin((πt)/4), et
2−2t) for
0 ≤ t ≤ 2
(a) 16
(b) 0
(c) −8
(d) 2π
(e) −9
11. Let f(x, y, z) = 3x2yez, find Div(∇(f(−1, 1, 3))).
(a) 6e3
(b) 3e3
(c) e3
(d) 9e3
(e) 0
3
12. Find a normal vector to the surface of the paraboloid z = x2+y2 at the point (3, 1, 10).
(a) (−6, 2, 1)
(b) (6, 2,−1)
(c) (−6,−2,−1)
(d) (−3, 2,−1)
(e) (6,−2, 1)
13. Let f(x) =x2
4− z , calculate∫∫
f(x, y, z)ds where S is the surface given by z = 4 −x2 − y2 for 0 ≤ z ≤ 3.
(a) π/12(17√
17− 5√
5)
(b) π/12(17√
17 + 5√
5)
(c) π/6(17√
17− 5√
5)
(d) π2/12(17√
17− 5√
5)
(e) 0
14. Calculate the flux of the vector field F = 〈0, 3, x〉 through the surface defined by thepart of the sphere x2 + y2 + z2 = 9 where x, y, z ≥ 0 with outward pointing normal.
(a) 27/4(3π − 4)
(b) 27/12(3π − 4)
(c) 27/4(3π + 4)
(d) 27/12(3π − 12)
(e) 9/4(3π + 4)
15. Let S be the portion of the sphere x2 + y2 + z2 = 9 where 1 ≤ x2 + y2 ≤ 4 adn z ≥ 0.What is the area of S? (Hint: Use a surface integral)
(a) 6π(√
8 +√
5)
(b) 6π(√
2−√
5)
(c) 6π(√
8−√
5)
(d) 3π(√
8−√
5)
(e) 3π
4
16. Let S be a surface given by the graph of z = xey, x, y ∈ [0, 1] oriented so that the zcomponent of the normal vector is positive. If F = 〈xy, 4x2, xyey〉 the flux of F acrossS is equal to;
(a) −1− e(b) −1 + e
(c) 1− e(d) e+ 1
(e) 0
17. Let S be the annular region given by 1 ≤ x2 + y2 ≤ 4 and let F = 〈−x2y, xy2〉 bea vector field. Suppose the boundary of the inner circle is oriented clockwise and theboundary of the outer circle is oriented counterclockwise. Calculate
∮∂S F. dr.
(a) 0
(b) π2
(c) −π(d) 103
(e) 126.5π
18. Calculate the area of the region between the x axis and the cycloid parametrized byc(t) = 〈t− sin t, 1− cos t〉 for 0 ≤ t ≤ 2π
(a) 0
(b) 6π
(c) 3π
(d) π
(e) 3π2
19. Calculate
∫∫
SzdS where S is the part of the surface z = x3 where x, y ∈ [0, 1].
(a)5√
10
27− 1
54
(b)5√
10
9+
1
54
(c)5√
10
27+
5
54
(d)5√
10
27+
1
54
(e)5√
15
27− 1
54
5
20. Let F = 〈x2 − y2, 2xy, z〉 and S be the surface that bounds the ellipsoid 9x2 + 16y2 +4z2 = 36. The flux of Curl(F) over S is given by
(a) 0
(b) π
(c) −π(d) 2π
(e) 9π2
21. Let F = 〈−z2, 2zx, 4y − xr〉 and Let C be a simple closed curve bounded by the planex+ y + z = 4 in the first octant, and suppose C is counterclockwise when viewed fromabove.
The value of
∫
CF.dr is given by;
(a) 32
(b) 16
(c) 8
(d) 4
(e) 2
22. Let F be a vector filed with differentiable component functions. If Curl(F)=0 in aconnected domain, then F is conservative
(a) TRUE
(b) FALSE
23. Let F = 〈z,−z, x2 − y2〉, let C consist of the three line segments that bound the planez = 8 − 4x − 2y in the first octant, and suppose C is counterclockwise when viewed
from above. Find The value of
∫
CF.dr.
(a) 0
(b) −88
3
(c) −71
6
(d) −23
3
(e) −71
3
6
Answer Key
Multiple Choice Questions
1. B2. C3. D4. E5. C6. B7. B8. D9. C10. A11. D12. B13. A14. E15. C16. A17. E18. B19. D20. A21. A22. B
7
