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Year 1 > Geometry & Motion > Flashcards

Geometry & Motion Flashcards

1
Q

Define a vector function

A
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2
Q

What is the parameterisation of a circle with centre (0,0) and radius R

A

r(t) = (Rcost, Rsint)

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3
Q

What is the parameterisation of a circle with centre (a,b) and radius R

A

r(t) = (a + Rcost, b + Rsint)

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4
Q

What is the parameterisation of a parabola

A

r(t) = (t,at2)

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5
Q

What is the parameterisation of the top half of a hyperbola

A

r(t) = (bsinht, acosht)

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6
Q

When is a curve closed

A

if I = [a,b] then r(a) = r(b)

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7
Q

When is a curve embedded

A

If it doesnt intersect

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8
Q

What is the general equation for an ellipse?

A

(x/a)2 + (y/b)2 = 1

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9
Q

What is the general equation for a hyperbola

A

(x/a)2 - (y/b)2 = 1

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10
Q

Find a parameterisation of the curve traced by a marked point on the perimeter of a unit circle that is rolling counter clockwise and without slippage around a fixed unit circle centred at the origin

A
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11
Q

What is the length of the curve l(c) denoted as

A
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12
Q

When is a parameterisation a natural parameterisation (or arc-length)

A

When the absolute value of the derivative is 1

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13
Q

Let r(t) be a parameterisation of a curve C in R3 such that r(0) = (R,-R,R). Suppose r(t) doesnt equal zero and r(t) . r’(t) = 0 for all points of C. Show that any such C must lie on the surface of a sphere. Find the position of the sphere’s centre and determine its radius

A
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14
Q

In two dimensions, mark a point on the outer rim of a wheel of radius R0 rolling on the horizonal line y=0 without slippage. Assume the centre of the wheel is moving with velocity v=V0i and that the marked point is (0,0) at time t=0.

i) Find r(t)
ii) compute the path between two consecutive points where the path touches y=0

A
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15
Q

How would you compute the length of the segment of the cardioid (r, theta) = (2 - 2cost, t)

A

Convert to cartesian, then l(c) = integral of the absolute value of the derivative

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16
Q
A
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17
Q

Give the standard integral results of these functions

A
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18
Q

Give the equation for the unit tangent vector

A

T = r’ / ||r’||

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19
Q

Give the equation for the principal normal vector

A

N = T’ / ||T’||

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20
Q

What is the equation for curvature, and the radius of curvature?

A

k = ||dT/ds||, p=1/k

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21
Q

What is the equation for curvature in terms of T and r

A

k = ||T’|| / ||r’||

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22
Q

What is the equation for curvature containing r and its derivatives?

A

k = ||r’(t) x r’‘(t)|| / ||r’(t)||3

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23
Q

Give the two Frenet-Serret equations

A

T’(s) = k(s)N(s)

N’(s) = -k(s)T’(s)

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24
Q

Define the Binormal Vector

A

B = T x N

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25
Q

Give the equation for torsion t

A

|t| = ||dB/ds||

t = (N . B’)/ ||r’||

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26
Q

What is the radius of an osculating circle

A

1/k

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27
Q

Use Frenet coordinates to describe all planar curves with constant curvature k>0

A
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28
Q

Describe all curves in Rn with constant zero curvature

A
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29
Q

Define a function of several variables

A

f: U in Rn to R is a rule that assigns a real number to each point in U

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30
Q

What is a level set Lk

A

A set of points for a function f: U in Rn to R such that

Lk = { x in U ; f(x) = k}

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31
Q

What is the gradient vector function?

A
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32
Q

State the chain rule

A
33
Q

Define the directional derivative in the direction of u

A

Duf(x) = grad f(x) . u

34
Q
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35
Q
A
36
Q
A
37
Q

Define the directional derivative including theta

A

Duf(x) = || grad f(x) || cos(theta)

38
Q

Find all points at which the direction of steepest descent of the function f(x,y) = x2 + y2 - 2x - 4y is in the direction (1/sqrt(2), 1/sqrt(2))

A
39
Q

What is the direction of steepest descent?

A
  • grad f
40
Q
A
41
Q

When is a point (a,b) a stationary point of f(x,y)

A

when both partial derivatives are zero

42
Q

Whats the second derivative test and give its 4 results

A

D = d2f/dx2(x,y)d2f/dy2(a,b) - [d2f/dxdy(ab)]2

if D > 0 and d2f/dx2(a,b) > 0 then f is a local minimum

if D > 0 and d2f/dx2(a,b) < 0 then f is a local maximum

if D < 0 then f(a,b) is a saddle point

if D = 0 then it’s inconclusive

43
Q

Two surfaces are called orthogonal at a point of intersection if their normals are perpendicular at that point. Show that the round cone z2 = x2 + y2 and the unit sphere x2 + y2 + z2 = 1 are orthogonal at every point of intersection

A
44
Q

What is the double integral of f over the rectangle R in terms of summations

A
45
Q

When is the double integral seperate?

A

When the function f(x,y)=g(x)h(y)

46
Q

What is the double integral of f over a rectangular box in terms of summations

A
47
Q

What is the mass of a solid M in terms of integrals

A

The triple integral of p, the density

48
Q

Give the centre of mass equations

A
49
Q
A
50
Q
A
51
Q

Consider the solid of constant density p bounded by the parabolic cylinder x=y2 and the planes x=z, z=0 and x=1. Write down an expression for the total mass as a 3d integral

A
52
Q

Write down the relationships between polar coordinates (r,theta) and cartesian coordinates (x,y)

A

x = rcos(theta) y= rsin(theta) x2+y2=r2 tan(theta) = y/x

53
Q

Write the integral of a rectangle in polar coordinates, specifically what dA is equal to

A
54
Q

Write down the relationships between cylindrical coordinates (r, theta, z) and cartesian coordinates (x,y,z)

A

x = rcos(theta) y=rsin(theta) x2+y2=r2 tan(theta)=y/x z=z

55
Q

Write down the volume of a rectangular box in cylindrical coordinates, specifically what dV is equal to

A
56
Q

Write down the relationship between spherical coordinates (r, theta, phi) and cartesian coordinates (x,y,z) and state the ranges of r, theta and phi

A

x=r cos(theta) sin(phi)

y = r sin(theta) cos(phi)

z = r cos(phi)

r2 = x2 + y2 + z2

r = [0,inf) phi = [0,pi] theta = [0,2pi]

57
Q

Write down the volume of a rectangular box in spherical coordinates, specifically what dV is equal to

A
58
Q
A
59
Q
A
60
Q
A
61
Q
A
62
Q

Define dA for a linear transformation

A

dA = |J| du dv

J is the jacobian of the partial derivatives so |J| is the determinant

63
Q

What is the double integral for a conversion of a linear transformation

A
64
Q

What is the Jacobian matrix d(x,y)/d(u,v)

A
65
Q

What is the jacobian matrix d(x,y,z)/d(u,v,w)

A
66
Q
A
67
Q
A
68
Q
A
69
Q

Define a parametric surface

A

S = {r(u,v) | (u,v) in Omega}

70
Q

What is the parameterisation for a unit sphere

A

r(u,v) = (cosu sinv, sinu sinv, cos v)

71
Q

What is the parameterisation for a cylinder of radius R

A

r(theta, t) = (Rcos(theta), Rsin(theta), t)

72
Q

What is the parameterisation for a Torus?

A

r(u,v) = ((a +bcosu)cosv, (a+bcosu)sinv, bsinu)

73
Q

What is the equation for the tangent plane at the (u0, v0)?

A

p(h,k) = r(u0, v0) + h(dr/du)(u0, v0) + k(dr/dv)(u0, v0) where h,k are in R2

74
Q

What is the equation for the unit normal vectors and the equation for the tangent plane including the normal vector

A
75
Q

What is the area of S as a double integral?

A
76
Q

For a parameterisation of a surface S what is the surface integral of f over S?

A
77
Q

What is the flux of v through a surface S?

A
78
Q
A

Year 1 (11 decks)

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