Geometry & Motion

Question 1

Define a vector function

Answer 1

Question 2

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

Answer 2

r(t) = (Rcost, Rsint)

Question 3

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

Answer 3

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

Question 4

What is the parameterisation of a parabola

Answer 4

r(t) = (t,at²)

Question 5

What is the parameterisation of the top half of a hyperbola

Answer 5

r(t) = (bsinht, acosht)

Question 6

When is a curve closed

Answer 6

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

Question 7

When is a curve embedded

Answer 7

If it doesnt intersect

Question 8

What is the general equation for an ellipse?

Answer 8

(x/a)² + (y/b)² = 1

Question 9

What is the general equation for a hyperbola

Answer 9

(x/a)² - (y/b)² = 1

Question 10

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

Answer 10

Question 11

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

Answer 11

Question 12

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

Answer 12

When the absolute value of the derivative is 1

Question 13

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

Answer 13

Question 14

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

Answer 14

Question 15

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

Answer 15

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

Question 16

Answer 16

Question 17

Give the standard integral results of these functions

Answer 17

Question 18

Give the equation for the unit tangent vector

Answer 18

T = r’ / ||r’||

Question 19

Give the equation for the principal normal vector

Answer 19

N = T’ / ||T’||

Question 20

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

Answer 20

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

Question 21

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

Answer 21

k = ||T’|| / ||r’||

Question 22

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

Answer 22

k = ||r’(t) x r’‘(t)|| / ||r’(t)||³

Question 23

Give the two Frenet-Serret equations

Answer 23

T’(s) = k(s)N(s)

N’(s) = -k(s)T’(s)

Question 24

Define the Binormal Vector

Answer 24

B = T x N

Question 25

Give the equation for torsion t

Answer 25

|t| = ||dB/ds|| t = (N . B')/ ||r'||

Question 26

What is the radius of an osculating circle

Answer 26

1/k

Question 27

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

Answer 27

Question 28

Describe all curves in Rⁿ with constant zero curvature

Answer 28

Question 29

Define a function of several variables

Answer 29

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

Question 30

What is a level set L_k

Answer 30

A set of points for a function f: U in Rⁿ to R such that L_k = { x in U ; f(x) = k}

Question 31

What is the gradient vector function?

Answer 31

Question 32

State the chain rule

Answer 32

Question 33

Define the directional derivative in the direction of u

Answer 33

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

Question 34

Answer 34

Question 35

Answer 35

Question 36

Answer 36

Question 37

Define the directional derivative including theta

Answer 37

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

Question 38

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))

Answer 38

Question 39

What is the direction of steepest descent?

Answer 39

- grad f

Question 40

Answer 40

Question 41

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

Answer 41

when both partial derivatives are zero

Question 42

Whats the second derivative test and give its 4 results

Answer 42

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

Question 43

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

Answer 43

Question 44

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

Answer 44

Question 45

When is the double integral seperate?

Answer 45

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

Question 46

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

Answer 46

Question 47

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

Answer 47

The triple integral of p, the density

Question 48

Give the centre of mass equations

Answer 48

Question 49

Answer 49

Question 50

Answer 50

Question 51

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

Answer 51

Question 52

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

Answer 52

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

Question 53

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

Answer 53

Question 54

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

Answer 54

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

Question 55

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

Answer 55

Question 56

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

Answer 56

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]

Question 57

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

Answer 57

Question 58

Answer 58

Question 59

Answer 59

Question 60

Answer 60

Question 61

Answer 61

Question 62

Define dA for a linear transformation

Answer 62

dA = |J| du dv J is the jacobian of the partial derivatives so |J| is the determinant

Question 63

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

Answer 63

Question 64

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

Answer 64

Question 65

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

Answer 65

Question 66

Answer 66

Question 67

Answer 67

Question 68

Answer 68

Question 69

Define a parametric surface

Answer 69

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

Question 70

What is the parameterisation for a unit sphere

Answer 70

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

Question 71

What is the parameterisation for a cylinder of radius R

Answer 71

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

Question 72

What is the parameterisation for a Torus?

Answer 72

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

Question 73

What is the equation for the tangent plane at the (u_0, v₀)?

Answer 73

p(h,k) = r(u_0, v₀) + h(dr/du)(u_0, v₀) + k(dr/dv)(u_0, v₀) where h,k are in R²

Question 74

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

Answer 74

Question 75

What is the area of S as a double integral?

Answer 75

Question 76

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

Answer 76

Question 77

What is the flux of v through a surface S?

Answer 77

Question 78

Answer 78