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Year 1 - Mathematics > Geometry > Flashcards

Flashcards in Geometry Deck (130)
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1
Q

What is a vector?

A

A list of real numbers

2
Q

What does Rⁿ mean?

A

For a given n, we denote the set of all vectors with n co-ordinates as Rⁿ, and
often refer to Rⁿ as n-dimensional co-ordinate space or simply as n-dimensional space

3
Q

What is the standard/canconical basis of Rⁿ

A
e₁, e₂, ..., eₙ
(1, 0, ..., 0)
(0, 1, ...., 0)
....
(0, 0, ...., 1)
4
Q

What is the triangle inequality?

A

u + v | ≤ |u| + |v|

5
Q

Prove the triangle inequality

A

Pg7

6
Q

What is the dot/scalar/Euclidean inner product?

A

u . v = u₁v₁ + u₂v₂ + … + uₙvₙ

7
Q

Let u, v, w be vectors in Rn and let λ be a real number. Then
(a) commutivity
(b) (λu) · v = [ ].
(c) (u + v) · w = [ ].
(d) [ ] . [ ] = |u|² ≥ 0 and u · u = 0 if and only if [ ].
(e) Cauchy-Schwarz Inequality
|u · v| ≤ |u| |v| (1.3)
with equality when one of u and v is a multiple of the other.

A
Let u, v, w be vectors in Rn and let λ be a real number. Then
(a) u · v = v · u.
(b) (λu) · v = λ(u · v).
(c) (u + v) · w = u · w + v · w.
(d) u · u = |u|
2  0 and u · u = 0 if and only if u = 0.
(e) Cauchy-Schwarz Inequality
|u · v|  |u| |v| (1.3)
with equality when one of u and v is a multiple of the other.
8
Q

|u| = ??

In terms of dot product

A

|u| = √ u.u

9
Q

What is the angle between two vectors u and v?

A

cos⁻¹(u.v/(|u| |v|))

10
Q

two vectors u and v are perpendicular if and only if [ ] = 0

A

u . v = 0

11
Q

What is the cosine rule?

A

a² = b² + c² - 2bccosα

12
Q

Prove the cosine rule

A

proof pg 9

13
Q

What is Thales theorem?

A

Let A and B be points at opposite ends of the diameter of a circle, and let P be a third point. Then ∡AP B is a right angle if and only if P also lies on the circle.

14
Q

Prove Thales theorem

A

Pg 9 (end)

15
Q

The medians of a triangle are concurrent at its [ ]

A

Centroid

16
Q

Prove:

The medians of a triangle are concurrent at its centroid

A

pg10

17
Q

Describe the parametric form of a line

A

Let p, a be vectors in Rⁿ with a ≠ 0. Then the equation r(λ) = p + λa, where λ is a real number, is the equation of the line through p, parallel to a. It is said to be in parametric form, the parameter here being λ. The parameter acts as a co-ordinate on the line, uniquely associating to each point on the line a value of λ.

18
Q

When are two vectors linearly independent?

A

We say that two vectors in Rⁿ are linearly independent, or just simply independent, if neither is a scalar multiple of the other.
In particular, this means that both vectors are non-zero.

19
Q

Two vectors which aren’t independent are said to be [ ]

A

linearly dependent

20
Q

What is the parametric form of a plane?

A

Let p, a, b be vectors in Rⁿ with a, b independent.
Then
r(λ, µ) = p + λa + µb where λ, µ are real numbers is the equation of the plane through p parallel to the vectors a, b. The parameters λ, µ act as co-ordinates in the plane, associating to each point of the plane a unique ordered pair (λ, µ) for if
p + λ₁a + µ₁b = p + λ₂a + µ₂b
then (λ₁ − λ₂) a = (µ₂ − µ₁
) b so that λ₁ = λ₂ and µ₁ = µ₂
by independence

21
Q

What is the Cartesian Equation of a Plane in R³?

A

A region Π of R³ is a plane if
and only if it can be written as r · n = c
where r = (x, y, z), n = (n₁, n₂, n₃) ≠ 0 and c is a real number. In terms of the co-ordinates
x, y, z this equation reads
n₁x + n₂y + n₃z = c
The vector n is normal (i.e. perpendicular) to the plane

22
Q

Prove the Cartesian Equation of the plane in R³

A

Proof pg 14

23
Q

What is the vector/cross product?

A

u ∧ v =
| i j k |
| u₁ u₂ u₃|
| v₁ v₂ v₃ |

24
Q

For u, v in R³, we have

|u ∧ v|² =

A

|u ∧ v|² = |u|²|v|² - (u.v)²

25
Q

When does u ∧ v = 0?

A

In particular u ∧ v = 0 if and only if u and v are linearly dependent.

26
Q

|u ∧ v| = |u| |v| [ ]

A

For u, v in R³ we have |u ∧ v| = |u| |v|sin θ where θ is the smaller angle between u and v

27
Q

Prove that |u ∧ v| = |u| |v|sin θ

A

|u ∧ v|² = |u|²|v|² - (u.v)² = |u|²|v|²(1 - cos²θ) = |u|²|v|² sin²θ

28
Q

For u, v in R³ then |u ∧ v| equals the area of the parallelogram with vertices….

A

0, u, v, u+v

29
Q

For u, v, w in R³, and reals α, β we have:

(αu + βv) ∧ w = [ ]

A

(αu + βv) ∧ w = α(u ∧ w) + β(v ∧ w)

30
Q

For u, v, w in R³, and reals α, β we have:

u ∧ v = − [ ]

A

u ∧ v = −v ∧ u.

31
Q

For u, v, w in R³, and reals α, β we have:

u ∧ v is perpendicular to [ ]

A

u ∧ v is perpendicular to both u and v

32
Q

For u, v, w in R³, and reals α, β we have:

If u, v are perpendicular unit vectors then [ ] is a unit vector

A

If u, v are perpendicular unit vectors then u ∧ v is a unit vector.

33
Q

For u, v, w in R³, and reals α, β we have:

If u, v are perpendicular unit vectors then [ ] is a unit vector

A

If u, v are perpendicular unit vectors then u ∧ v is a unit vector.

34
Q

Given independent vectors u and v in R³, the plane containing the origin 0 and
parallel to u and v has equation ….

A

n r · (u ∧ v) = 0

35
Q

Given independent vectors u and v in R³, the plane containing the origin 0 and
parallel to u and v has equation ….

A

r · (u ∧ v) = 0

36
Q

Define the scalar triple product

[u, v, w]

A

[u, v, w] = u . ( v ∧ w)

37
Q

[u, v, w] = [ ] [v, w, u] = [ ] [w, u, v] = [ ] [u, w, v] = [ ] [v, u, w] = [ ] [w, v, u]

A

[u, v, w] = [v, w, u] = [w, u, v] = − [u, w, v] = − [v, u, w] = − [w, v, u]

38
Q

Note that [u, v, w] = 0 if and only if ….

A

Note that [u, v, w] = 0 if and only if u, v, w are linearly dependent; this is equivalent to the 3 × 3 matrix with rows u, v, w being singular.

39
Q

What is the volume of a parallelepiped with vertices 0, u, v, w?

A

|[u, v, w]|

40
Q

What is the vector triple product?

A

Given three vectors u, v, w in R³ we define their vector triple product as
u ∧ (v ∧ w).

41
Q

u ∧ (v ∧ w) = ??? (In terms of dot and vector product)

A

u ∧ (v ∧ w) = (u · w)v − (u · v)w

42
Q

Prove that u ∧ (v ∧ w) = (u · w)v − (u · v)w

A

proof end pg20

43
Q

What is the scalar quadruple product?

A

Given four vectors a, b, c, d in R³, their scalar quadruple product is
(a ∧ b) · (c ∧ d)

44
Q

(a ∧ b) · (c ∧ d) = ??? (In terms of dot product)

A

(a ∧ b) · (c ∧ d) = (a · c)(b · d) − (a · d)(b · c)

45
Q

Prove that

a ∧ b) · (c ∧ d) = (a · c)(b · d) − (a · d)(b · c

A
Set e = c ∧ d. Then
(a ∧ b) · (c ∧ d)
= e · (a ∧ b)
= [e, a, b]
= [a, b, e] 
= a · (b ∧ e)
= a · (b ∧ (c ∧ d))
= a · ((b · d)c − (b · c)d) [by the vector triple product]
= (a · c)(b · d) − (a · d)(b · c).
46
Q

Let a and b be linearly independent vectors in R³. Then a, b and a∧b form
a basis for R³. This means that for every v in R³ there are unique real numbers α, β, γ such
that v = [ ]
We will refer to α, β, γ as the co-ordinates of v with respect to this basis

A

v = αa + βb + γa ∧ b

47
Q

What is the vector equation of a line using vector product?

A

Let a, b vectors in R³ with a · b = 0 and a ≠ 0. The vectors r in R³ which satisfy r ∧ a = b form the line parallel to a which passes
through the point (a ∧ b)/ |a|²

48
Q

det AB =

A

det AB = det A det B

49
Q

a square matrix is singular if and only if [ ].

A

a square matrix is singular if and only if has zero determinant.

50
Q

What is the equation for a double cone?

A

x² + y² = z²cot²α

51
Q

What are the four types of conic sections?

A

Circle
Ellipse
Parabola
Hyperbola

52
Q

Let D be a line, F a point not on the line D and e > 0. Then the conic with directrix D and focus F and eccentricity e, is the set of points P (in the plane containing
F and D) which satisfy the equation ….

A

|P F| = e|PD|
where |P F| is the distance of P from the focus and |PD| is the distance of P from the directrix.
That is, as the point P moves around the conic, the shortest distance from P to the line D is in constant proportion to the distance of P from the point F.

53
Q

Conics

If 0 < e < 1 then the conic is called an [ ]

A

ellipse

54
Q

Conics

If e = 1 then the conic is called a [ ]

A

parabola

55
Q

Conics

If e > 1 then the conic is called a [ ]

A

hyperbola

56
Q

What is the equation of an ellipse? Define each variable

xy-co-ordinates

A

x²/a² + y²/b² = 1 0 < b < a
where
a = ke/(1-e²) and b = ke/√ (1-e²)

and e = √ (1 - b²/a²)

57
Q

In the xy-co-ordinates,

where is the focus and directrix of an ellipse?

A

The focus F is at (ae, 0) and the directrix D is the line x = a/e

58
Q

What is the area of an ellipse in xy-co-ordinates?

A

πab

a = ke/(1-e²) and b = ke/√ (1-e²)
and e = √ (1 - b²/a²)

59
Q

How can an ellipse be parametrized using trig?

A

x = a cost, y = b sin t, 0 ≤ t < 2π

60
Q

How else can an ellipse be parametrized (no trig)?

A

x = a(1 - t²)/(1 + t²)
y = b(2t)/(1 + t²)
(−∞ < t < ∞)

61
Q

What is the equation of a hyperbola? Define each variable

xy-co-ordinates

A

x²/a² - y²/b² = 1 0 < a, b
where
a = ke/(e² - 1) and b = ke/√ (e² - 1)

and e = √ (1 + b²/a²)

62
Q

In the xy-co-ordinates,

where is the focus and directrix of a hyperbola?

A

The focus F is at (ae, 0) and the directrix D is the line x = a/e

63
Q

What are the equations for the asymptotes of a hyperbola?

A

The lines ay = ±bx are known as the asymptotes of the hyperbola; these are, in a sense, the tangents to the hyperbola at its two ‘points at infinity’.

64
Q

What is a right hyperbola?

A

When e = √2 (i.e. when a = b) then the asymptotes are perpendicular and C is known as a right hyperbola.

65
Q

How can a hyperbola be parametrized using hyperbolic trig?

A

x = ±a cosh t y = b sinh t

−∞ < t < ∞)

66
Q

How else can a hyperbola be parametrized (no trig)?

A

x = a(1 + t²)/(1 - t²)
y = b(2t)/(1 - t²)
(t ≠ ±1)

67
Q

What is the normal form of a parabola?

A

y² = 4ax

68
Q

In the xy-co-ordinates,

where is the focus and directrix of a parabola?

A

The focus is the point (a, 0) and the directrix is the

line x = −a

69
Q

Where is the vertex of a parabola?

A

The vertex of the parabola is at (0, 0)

70
Q

What is the parametrized form of a parabola?

A

(x, y) = (at², 2at) where −∞ < t < ∞.

71
Q

What is the eccentricity of a circle?

A

e = 0

72
Q

Let A, B be distinct points in the plane and r > |AB| be a real number. The
locus |AP| + |P B| = r is an [ ] with foci at [ ] and [ ]

A

Let A, B be distinct points in the plane and r > |AB| be a real number. The
locus |AP| + |P B| = r is an ellipse with foci at A and B

73
Q

Let A, B be distinct points in the plane and r > |AB| be a real number. The
locus |AP| + |P B| = r is an ellipse with foci at A and B

prove it

A

If we consider the ellipse with foci A and B and a point P on the ellipse we have
|AP| + |P B| = |F₁P| + |P F₂|
= e |D₁P| + e |D₂P|
= e |D₁D₂|
where D₁D₂ is the perpendicular distance between the two directrices. Thus the value is constant on any ellipse with foci A and B and will take different values for different ellipses as the value of |AP| + |P B| increases as P moves right along the line AB.

74
Q

Show that the curve
x² + xy + y² = 1
is an ellipse

A

A rotation about the origin in R² by θ anti-clockwise takes the form
X = x cos θ + y sin θ, Y = −x sin θ + y cos θ;
x = X cos θ − Y sin θ, y = X sin θ + Y cos θ.
Writing c = cos θ and s = sin θ, for ease of notation, our equation becomes
(Xc − Y s)² + (Xc − Y s)(Xs + Y c) + (Xs + Y c)² = 1
which simplifies to
(1 + cs)X² + (c² − s²)XY + (1 − cs)Y²= 1.
So if we wish to eliminate the xy-term then we want
cos 2θ = c² − s²= 0
which will be the case when θ = π/4, say. For this value of θ we have c = s = 1/√2 and our equation has become
3/2 X² + 1/2 Y² = 1.

75
Q

The solutions of the equation
Ax² + Bxy + Cy² = 1
where A, B, C are real constants, such that A, B, C are not all zero, form one of the following
types of loci: ….

A

pes of loci:
Case (a): If B² − 4AC < 0 then the solutions form an ellipse or the empty set.
Case (b): If B² − 4AC = 0 then the solutions form two parallel lines or the empty set.
Case (c): If B² − 4AC > 0 then the solutions form a hyperbola.

76
Q

Ax² + Bxy + Cy² + Dx + Ey + F = 0

Describe the different cases for the loci of solutions

A

Case (a): If B² − 4AC < 0 then the solutions form an ellipse, a single point or the empty set.
Case (b): If B² − 4AC = 0 then the solutions form a parabola, two parallel lines, ansingle line or the empty set.
Case (c): If B² − 4AC > 0 then the solutions form a hyperbola or two intersecting
lines.

77
Q

What is an isometry?

A

An isometry T from Rⁿ to Rⁿ is a distance-preserving map. That is:
|T(x) − T(y)| = |x − y| for all x, y in Rⁿ

78
Q

Rotations, [ ] and [ ] are all examples of isometries

A

reflections

translations

79
Q

Describe the map Rθ, which denotes rotation by θ anti-clockwise about the origin

A

Rθ = ( cosθ - sinθ )

( sinθ cosθ )

80
Q

Describe the map Sθ, which is reflection in the line

y = x tan θ

A

Sθ = ( cos2θ sin2θ )

( sin2θ -cos2θ )

81
Q

A square real matrix A is said to be orthogonal if ….

A

A⁻¹ = Aᵀ

82
Q

The n × n orthogonal matrices are the linear [ ] of Rⁿ

A

isometries

83
Q

Let A be an n × n orthogonal matrix and b ∈ Rⁿ. The map
T(x) = Ax + b
is an [ ] of Rⁿ

A

isometry

84
Q

Let A and B be n × n orthogonal matrices

Is AB orthogonal?

A

Yes

85
Q

Let A and B be n × n orthogonal matrices

Is A⁻¹ orthogonal?

A

Yes

86
Q

Let A and B be n × n orthogonal matrices

What is detA = ???

A

detA = ±1

87
Q

We say that n vectors v₁, v₂, …, vₙ are a basis for Rⁿ if …

A

if for every v ∈ Rⁿ there exist unique real numbers α₁, α₂, …, αₙ such that
v = α₁v₁ + α₂v₂ + · · · + αₙvₙ.

88
Q

What is an orthonormal basis?

A

A basis v₁, v₂, …, vₙ for Rⁿ is orthonormal if
vᵢ · vⱼ = δᵢⱼ = { 1 if i = j
{ 0 if i ≠ j

89
Q

[ ] orthonormal vectors in Rⁿ

form a basis. (In fact, orthogonality alone is sufficient to guarantee linear [ ] .)

A

n

independence

90
Q

If v₁, v₂, …, vₙ are an orthonormal basis for Rⁿ and x = α₁v₁ + α₂v₂ + · · · + αₙvₙ
then note that
αᵢ = [ ]

A

αᵢ = x · vᵢ

91
Q

An n × n matrix A is orthogonal if and only if …

A

An n × n matrix A is orthogonal if and only if its columns form an orthonormal basis for Rⁿ. The same result hold true for the rows of A.

92
Q

Let A be an orthogonal 3 × 3 matrix, and x, y be column vectors in R³.

(a) If det A = 1 then A(x ∧ y) = [ ]
(b) If det A = −1 then A(x ∧ y) = [ ]

A

Let A be an orthogonal 3 × 3 matrix, and x, y be column vectors in R³

(a) If det A = 1 then A(x ∧ y) = Ax ∧ Ay.
(b) If det A = −1 then A(x ∧ y) = −Ax ∧ Ay.

93
Q

What is the Spectral Theorem (Finite dimensional case)?

A

Let A be a square real symmetric matrix (so Aᵀ = A). Then there is an orthogonal matrix P such that PᵀAP is diagonal.

94
Q

(Classifying 3 × 3 orthogonal matrices) Let A be a 3 ×3 orthogonal matrix.
If det A = 1 then A is a ….

A

If det A = 1 then A is a rotation of R³ about some axis by an angle θ where
trace A = 1 + 2 cos θ.

95
Q

(Classifying 3 × 3 orthogonal matrices) Let A be a 3 ×3 orthogonal matrix.
If det A = −1 and trace A = 1 ….

A

If det A = −1 and trace A = 1 then A is a reflection of R³. The converse also holds.

96
Q

Let S be an isometry from Rn
to Rⁿ such that S(0) = 0. Then
(a) |S(v)| = [ ] for any v in Rⁿ and [ ] = u · v for any u, v in Rⁿ.
(b) If v1, …, vn is an orthonormal basis for Rⁿ then so is [ ]
(c) There exists an orthogonal matrix A such that S(v) = [ ] for each v in Rⁿ

A

Let S be an isometry from Rn
to Rⁿ such that S(0) = 0. Then
(a) |S(v)| = |v| for any v in Rⁿ and S(u) · S(v) = u · v for any u, v in Rⁿ.
(b) If v1, …, vn is an orthonormal basis for Rⁿ then so is S(v1), …, S(vn).
(c) There exists an orthogonal matrix A such that S(v) = Av for each v in Rⁿ

97
Q

Let T be an isometry from Rn
to Rⁿ. Then there is a orthogonal matrix A and a column vector b such that T(v) = [ ] for all v.
Further A and b are [ ] in this regard.

A

Av + b

unique

98
Q

Euler Angles
Let R denote an orthogonal 3 × 3 matrix with det R = 1 and
let
R (i, θ) =

A

R (i, θ) = ( 1 0 0 )
( 0 cosθ - sinθ )
( 0 sinθ cosθ )

99
Q

Let R denote an orthogonal 3 × 3 matrix with det R = 1 and
let
R (j, θ) =

A

R (j, θ) = ( cosθ 0 -sinθ )
( 0 1 0 )
( sinθ 0 cosθ )

100
Q

Let’s suppose that the fixed-in-body and fixed-in-space axes share a common origin throughout the motion. Then at a time t the moving axes will be a rotation A(t) from the fixed axes.
We then have the equation
A(t)A(t)ᵀ = I

What does
dA/dt Aᵀ = ???

A

dA/dt Aᵀ = (0 −γ β )
(γ 0 −α )
(−β α 0 )

for some α, β, γ ∈ R (which of course may still depend on t)

101
Q

Let’s suppose that the fixed-in-body and fixed-in-space axes share a common origin throughout the motion. Then at a time t the moving axes will be a rotation A(t) from the fixed axes.
We then have the equation
A(t)A(t)ᵀ = I
Define the angular velocity of a body at time t

A

ω(t) = (α, β, γ)ᵀ

102
Q

What does
( x )
ω ∧ ( y ) = ???
( z )

A

( x )
ω ∧ ( y ) = dA/dt Aᵀ (x, y, z)ᵀ
( z )

103
Q

Rotating frames
Let r(t) be the position vector at time t of a point fixed in the body — this is the point’s
position vector relative to fixed-in-space axes.
Then r(t) = A(t) [ ]
Differentiating this we
find ….

A
Then r(t) = A(t)r(0)
dr/dt = dA/dr r(0) = (dA/dt  Aᵀ) (Ar(0)) = 
  (0  −γ    β  )
  (γ    0  −α )   r(t)
  (−β  α    0 )

= ω(t) ∧ r(t)

104
Q

Give the Cartesian equation for the following parametrized surfaces in R³:
Sphere

A

x² + y² + z² = a²

105
Q

Give the Cartesian equation for the following parametrized surfaces in R³:
Ellipsoid

A

x²/a² + y²/b² + z²/c²

106
Q

Give the Cartesian equation for the following parametrized surfaces in R³:
Hyperboloid of Two Sheets

A

x²/a² - y²/b² - z²/c² = 1

107
Q

Give the Cartesian equation for the following parametrized surfaces in R³:
Hyperboloid of One Sheet

A

x²/a² + y²/b² - z²/c² = 1

108
Q

Give the Cartesian equation for the following parametrized surfaces in R³:
Paraboloid

A

z = x² + y²

109
Q

Give the Cartesian equation for the following parametrized surfaces in R³:
Hyperbolic Paraboloid

A

z = x² - y²

110
Q

Give the Cartesian equation for the following parametrized surfaces in R³:
Cone

A

z² = x² + y² with z ≥ 0

111
Q

Define arc length given a parametrized curve r: [a, b] → R³

A

arc length = ₐ∫ᵇ |r’(t)| dt

112
Q

What is a smooth parametrized surface?

A

A smooth parametrized surface is a map r, known as the parametrization
r : U → R³
given by r (u, v) = (x (u, v), y (u, v), z (u, v)) from an (open) subset U ⊆ R² to R³ such that:
• x, y, z have continuous partial derivatives with respect to u and v of all orders;
• r is a bijection from U to r(U) with both r and r⁻¹ being continuous;
• (smoothness condition) at each point the vectors
rᵤ = ∂r/∂u and rᵥ = ∂r/∂v
are linearly independent (i.e. are not scalar multiples of one another).

113
Q

Define Cylindrical Polar Co-ordinates)

A

These are given by
x = r cos θ, y = r sin θ, z = z,
with r > 0, −π < θ < π, z ∈ R.

114
Q

Define Spherical Polar Co-ordinates

A

These are given by
x = r sin θ cos φ, y = r sin θ sin φ, z = r cos θ,
with r > 0, 0 < θ < π, −π < φ < π.

115
Q

What is the parametrization of a cone using cylindrical

polar co-ordinates?

A

r(θ, z) = (z cos θ, z sin θ, z) 0 < θ < 2π, z > 0

Note that the parametrization misses one meridian of the cone

116
Q

Let r: U → R³ be a smooth parametrized surface and let p be a point on the
surface
What is the tangent plane to r(U) at p?

A

The plane containing p and which is parallel to the vectors
rᵤ(p) = ∂r/∂u (p)
rᵥ(p) = ∂r/∂v (p)

117
Q

Let r: U → R³ be a smooth parametrized surface and let p be a point on the
surface
Is the tangent plane to r(U) at p well defined? And why?

A

Because rᵤ and rᵥ are independent the tangent plane

is well-defined.

118
Q

Any vector in the direction
∂r/∂u (p) ∧ ∂r/∂v (p)
is said to be [ ] to the surface at p. Thus there are two [ ] of length one.

A

normal

unit normals

119
Q

What is the parametrization of a sphere using spherical polar co-ordinates?

A

Spherical polar co-ordinates give a natural parametrization for
the sphere x² + y² + z² = a² with
r (φ, θ) = (a sin θ cos φ, a sin θ sin φ, a cos θ), −π < φ < π, 0 < θ < π.

120
Q

We can form a surface of revolution by rotating the graph y = f(x), where f(x) > 0, about the x-axis. There is then a fairly natural parametrization for the surface of revolution with cylindrical polar co-ordinates:
r(x , θ) = ???

A

r(x, θ) = (x, f(x) cos θ, f(x) sin θ) − π < θ < π, a < x < b

121
Q

What is a catenoid and what is its parametrization?

A

The catenoid is the surface of revolution formed by rotating the curve y =
cosh x, known as a catenary, about the x-axis. So we can parametrize it as
r(x, θ) = (x, cosh x cos θ, cosh x sin θ), −π < θ < π, x ∈ R

122
Q

What is a helicoid and what is its parametrization?

A

The helicoid is formed in a “propeller-like” fashion by pushing the x-axis up the z-axis while
spinning the x-axis at a constant angular velocity. So we can parametrize it as
s(X, Z) = (X cosZ, X sinZ, Z)

123
Q

Let r : U → R³ be a smooth parametrized surface.

What is the (surface) area of r(U)?

A

∫∫ᵤ |∂r/∂u ∧ ∂r/∂v| du dv

124
Q

We will often write
dS = [ ] du dv
to denote an infinitesimal part of surface area

A

∂r/∂u ∧ ∂r/∂v|

125
Q

The surface area of r (U) is independent of the choice of [ ]

A

parametrization

126
Q

Let z = f (x, y) denote the graph of a function
f defined on a subset S of the xy-plane
What is the surface area of the graph?

A

∫∫ₛ√ (1 + (fₓ)² + (fᵧ)²) dx dy

127
Q

Let z = f (x, y) denote the graph of a function
f defined on a subset S of the xy-plane
Show that the graph has surface area ∫∫ₛ√ (1 + (fₓ)² + (fᵧ)²) dx dy

A
We can parametrize the surface as
r (x, y) = (x, y, f (x, y)) (x, y) ∈ S.
Then
rₓ ∧ rᵧ = (-fₓ, -fᵧ, 1)
Hence the graph has surface area 
∫∫ₛ |rₓ ∧ rᵧ| dx dy = ∫∫ₛ√ (1 + (fₓ)² + (fᵧ)²) dx dy
128
Q

A surface S is formed by rotating the graph of
y = f(x) a < x < b,
about the x-axis. (Here f(x) > 0 for all x.) The surface area of S equals ?????

A

Area(S) = 2π ₓ₌ₐ∫ˣ⁼ᵇ f(x) ds/dx dx

129
Q

Isometries preserve [ ]

A

Area

Angles (between curves)

130
Q

What is the volume of a tetrahedron with vertices 0, u, v, w?

A

1/6 |[u, v, w]|