Midterm 3 Review
Jordan Barrett, MAT 208
May 4th, 2023
”Why are numbers beautiful? It’s like asking why is Ludwig van Beethoven’s
Ninth Symphony beautiful.”
- P ál Erdős
1
The Calculus of Vector Valued Functions
1.1
Definitions and Useful Theorems
Definition 1 A vector valued function is a function whose input is a real
parameter t and whose output is a vector.
Theorem 1.1 Suppose ⃗r(t) = ⟨x(t), y(t), z(t)⟩ = x(t)⃗i + y(t)⃗j + z(t)⃗k. The
parametric derivative of r(t) is given by:
⃗r′ (t) = ⟨x′ (t), y ′ (t), z ′ (t)⟩
where x′ (t), y ′ (t) and z ′ (t) are the usual derivatives of a single variable function.
Theorem 1.2 Let f (t) be a real valued function of one variable and let ⃗r(t)
and ⃗s(t) be differentiable vector valued functions of the same variable. Then the
following hold:
+ ⃗s(t)] = ⃗r′ (t) + ⃗s′ (t)
1.
d
r(t)
dt [⃗
2.
d
r(t)]
dt [f (t)⃗
3.
d
s(t)
dt [⃗
· ⃗r(t)] = ⃗s′ (t) · ⃗r(t) + ⃗s(t) · ⃗r′ (t)
4.
d
s(t)
dt [⃗
× ⃗r(t)] = ⃗s′ (t) × ⃗r(t) + ⃗s(t) × ⃗r′ (t)
5.
d
r(f (t))]
dt [⃗
= f ′ (t)⃗r(t) + f (t)⃗r′ (t)
= f ′ (t)⃗r′ (f (t))
1
Definition 2 The integral of ⃗r(t) = ⟨x(t), y(t), z(t)⟩ with respect to t is given
by:
Z
Z
Z
Z
z(t)dt ⃗k
y(t)dt ⃗j +
x(t)dt ⃗i +
⃗r(t)dt =
1.2
Exercises
Exercise 1 Compute the derivatives of the following functions:
1. ⃗r(t) = ⟨ecos(t) , tsin(t2 ), ln(t)⟩
t
⟩
2. ⃗r(t) = ⟨t2 + 3t, e−2t , t2 +1
3
3. ⃗r(t) = ⟨t, t2 , t2t+t ⟩
4. ⃗r(t) = ⟨1, 2, 3⟩
Exercise 2 Compute the integrals of the following functions:
1. ⃗r(t) = ⟨et , tsin(t2 ), ln(t)⟩
t
⟩
2. ⃗r(t) = ⟨t2 + 3t, e−2t , t2 +1
3
3. ⃗r(t) = ⟨t, t2 , t2t+t ⟩
4. ⃗r(t) = ⟨4, 5, 6⟩
2
2
2.1
Vector Fields, Line Integrals and Parametric
Functions
Definitions and Useful Theorems
Definition 3 A vector field is a function which assigns to every point in
n-dimensional space an n-dimensional vector.
Theorem 2.1 Let C be a smooth, oriented curve traced out by the parameterization r(t), where a ≤ t ≤ b. If F is a continuous vector field containing C,
then the line integral of F along C is given by:
Z
Z
F · dr =
C
2.2
b
F (r(t)) · r′ (t)dt
a
Exercises
Exercise 3 Consider the two dimensional vector field given by F (x, y) = ⟨2x, −y⟩.
In the space below, draw two dimensional coordinate axes. Draw a sketch of
F (x, y) by picking several points and sketching the corresponding vectors.
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Exercise 4 Consider the curve C given by the parametric function
r(t) = ⟨2cos(t), sin(t) + 5⟩ with 0 ≤ t ≤ π2
1. Sketch this parametric curve
2. Compute the line integral of ⟨x + y, y 2 ⟩ along C.
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3
3.1
Path Independence, Curl, Green’s Theorem
Definitions and Useful Theorems
Definition 4 We say that a vector field F is path independent if for any two
oriented paths C1 and C2 beginning and ending at the same points we have:
Z
Z
F · d⃗r =
F · d⃗r
C1
C2
Theorem 3.1 Let f be a function for which ∇f is continuous on a domain D.
if P and Q are points in D and C is a piece-wise smooth, oriented path from P
to Q in D, then:
Z
∇f dr = f (Q) − f (P )
C
Definition 5 Let F = ⟨F1 .F2 , F3 ⟩ be a three dimensional vector field. The curl
of F is given by:
curl(F ) = ∇ × F
Theorem 3.2 (Green’s Theorem) Let C be a simple closed curve in the plane
that bounds a region R with C oriented in such a way that when walking along
C in the direction of its orientation, the region R is on our left. Suppose that
F = ⟨F1 , F2 ⟩ is vector field with continuous partial derivatives on the region R
and its boundary C. Then:
Z
Z Z ∂F1
∂F2
−
dA
F · dr =
∂X
∂Y
C
R
5
3.2
Exercises
Exercise 5 Suppose f (x, y) = 3xy 2 +ey . Compute the line integral of ∇f along
the unit circle.
Exercise 6 Suppose f (x, y) = 3xy 2 +ey . Compute the line integral of ∇f along
(x − h)2
(y − k)2
+
=1
2
b
a2
Exercise 7 Compute the curl F = ⟨x2 − y, y + 3z 2 , x3 ⟩. What is the curl of F
at (1, 1, 1)?
Exercise 8 Let C be the circle of radius 3 centered at (2, 1), oriented counterclockwise. Compute the line integral of F = ⟨y 2 , 5x + 2xy⟩ along C both directly
and using Green’s Theorem.
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4
4.1
Parametric Surfaces and the Flux of a Vector
Field Through a Surface
Definitions and Useful Theorems
Definition 6 Parametric surfaces are functions of the form:
r(s, t) = ⟨x(s, t), y(s, t), z(s, t)⟩
Definition 7 Let r(s, t) = ⟨x(s, t), y(s, t), z(s, t)⟩ be a parametrization of a
smooth surface. The area of the surface defined by r on a domain D is given by:
Z Z
|rs × rt |dA
D
Definition 8 Suppose r(s, t) parameterizes a smooth surface Q over a domain
D. The total flux of a smooth vector field F through Q is given by:
Z Z
F · (rs × rt )dA
D
Theorem 4.1 Give a surface of the form z = f (x, y) with parametrization
r(s, t) we have that
rs × rt = ⟨−
∂f
∂f
, − , 1⟩
∂x
∂y
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4.2
Exercises
Exercise 9 Consider the following parametric function:
r(s, t) = ⟨Rsin(s)cos(t), Rsin(s)sin(t), Rcos(s)⟩
Where R is a constant, 0 ≤ s ≤ π and 0 ≤ t ≤ 2π. What surface does this
parameterize?
Exercise 10 Compute the surface area of
r(s, t) = ⟨Rsin(s)cos(t), Rsin(s)sin(t), Rcos(s)⟩
Does your result look familiar?
Exercise 11 Compute the flux of F = ⟨x, y, z⟩ through the surface described in
exercise 9.
Exercise 12 Compute the flux of F = ⟨x2 , y, z⟩ through z = 2x3 + 3y 4 using
the theorem above.
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