Geometry & Motion Flashcards
Define a vector function
What is the parameterisation of a circle with centre (0,0) and radius R
r(t) = (Rcost, Rsint)
What is the parameterisation of a circle with centre (a,b) and radius R
r(t) = (a + Rcost, b + Rsint)
What is the parameterisation of a parabola
r(t) = (t,at2)
What is the parameterisation of the top half of a hyperbola
r(t) = (bsinht, acosht)
When is a curve closed
if I = [a,b] then r(a) = r(b)
When is a curve embedded
If it doesnt intersect
What is the general equation for an ellipse?
(x/a)2 + (y/b)2 = 1
What is the general equation for a hyperbola
(x/a)2 - (y/b)2 = 1
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
What is the length of the curve l(c) denoted as
When is a parameterisation a natural parameterisation (or arc-length)
When the absolute value of the derivative is 1
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
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
How would you compute the length of the segment of the cardioid (r, theta) = (2 - 2cost, t)
Convert to cartesian, then l(c) = integral of the absolute value of the derivative
Give the standard integral results of these functions
Give the equation for the unit tangent vector
T = r’ / ||r’||
Give the equation for the principal normal vector
N = T’ / ||T’||
What is the equation for curvature, and the radius of curvature?
k = ||dT/ds||, p=1/k
What is the equation for curvature in terms of T and r
k = ||T’|| / ||r’||
What is the equation for curvature containing r and its derivatives?
k = ||r’(t) x r’‘(t)|| / ||r’(t)||3
Give the two Frenet-Serret equations
T’(s) = k(s)N(s)
N’(s) = -k(s)T’(s)
Define the Binormal Vector
B = T x N
Give the equation for torsion t
|t| = ||dB/ds||
t = (N . B’)/ ||r’||
What is the radius of an osculating circle
1/k
Use Frenet coordinates to describe all planar curves with constant curvature k>0
Describe all curves in Rn with constant zero curvature
Define a function of several variables
f: U in Rn to R is a rule that assigns a real number to each point in U
What is a level set Lk
A set of points for a function f: U in Rn to R such that
Lk = { x in U ; f(x) = k}
What is the gradient vector function?
State the chain rule
Define the directional derivative in the direction of u
Duf(x) = grad f(x) . u
Define the directional derivative including theta
Duf(x) = || grad f(x) || cos(theta)
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))
What is the direction of steepest descent?
- grad f
When is a point (a,b) a stationary point of f(x,y)
when both partial derivatives are zero
Whats the second derivative test and give its 4 results
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
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
What is the double integral of f over the rectangle R in terms of summations
When is the double integral seperate?
When the function f(x,y)=g(x)h(y)
What is the double integral of f over a rectangular box in terms of summations
What is the mass of a solid M in terms of integrals
The triple integral of p, the density
Give the centre of mass equations
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
Write down the relationships between polar coordinates (r,theta) and cartesian coordinates (x,y)
x = rcos(theta) y= rsin(theta) x2+y2=r2 tan(theta) = y/x
Write the integral of a rectangle in polar coordinates, specifically what dA is equal to
Write down the relationships between cylindrical coordinates (r, theta, z) and cartesian coordinates (x,y,z)
x = rcos(theta) y=rsin(theta) x2+y2=r2 tan(theta)=y/x z=z
Write down the volume of a rectangular box in cylindrical coordinates, specifically what dV is equal to
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
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]
Write down the volume of a rectangular box in spherical coordinates, specifically what dV is equal to
Define dA for a linear transformation
dA = |J| du dv
J is the jacobian of the partial derivatives so |J| is the determinant
What is the double integral for a conversion of a linear transformation
What is the Jacobian matrix d(x,y)/d(u,v)
What is the jacobian matrix d(x,y,z)/d(u,v,w)
Define a parametric surface
S = {r(u,v) | (u,v) in Omega}
What is the parameterisation for a unit sphere
r(u,v) = (cosu sinv, sinu sinv, cos v)
What is the parameterisation for a cylinder of radius R
r(theta, t) = (Rcos(theta), Rsin(theta), t)
What is the parameterisation for a Torus?
r(u,v) = ((a +bcosu)cosv, (a+bcosu)sinv, bsinu)
What is the equation for the tangent plane at the (u0, v0)?
p(h,k) = r(u0, v0) + h(dr/du)(u0, v0) + k(dr/dv)(u0, v0) where h,k are in R2
What is the equation for the unit normal vectors and the equation for the tangent plane including the normal vector
What is the area of S as a double integral?
For a parameterisation of a surface S what is the surface integral of f over S?
What is the flux of v through a surface S?
