Practice Questions Flashcards

1
Q

How to find the eigenvalues and normalised eigenvectors of a matrix

A
  1. Solve the characteristic equation for A
  2. Solutions are the eigenvalues
  3. The eigenvector corresponding to the eigenvalue has to satisfy (A-λI)x(v) = 0

(v) is the vector

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2
Q

Fourier series for a square wave

A
  1. Fourier series expansion formula
  2. Find the coefficients
  3. odd -> cos is odd -> an = 0
  4. even -> sin is even -> bn = 0
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3
Q

Indirect calculation of Laplace transform

A
  1. Apply Laplace transform table
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4
Q

Inverse Laplace transform using partial fractions

A
  1. use partial fractions
  2. use table to solve
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5
Q

solution of ODE with constant coefficients using the Laplace transform

A
  1. Take the Laplace transform
  2. Substitute in the boundary conditions and rearrange
  3. Use partial fractions
  4. use table to solve
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6
Q

How to calculate the unit vector

A

a(v)(h) = a(v)/a

where a(v)(h) is the unit vector

and a = √(a(x)^2+a(y)^2)

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7
Q

how to calculate the projection of b onto the vector a

A
  1. the component of b that is in the direction of a
  2. evaluate the dot product a(hat) . b
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8
Q

what is the normal to the plane

A
  1. solve | r(v) . n(v)(h) | = d

where n(v) is the coefficients of the equation.
and r(v) = (x,y,z)

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9
Q

what is the shortest distance from plane

A
  1. | r(v) . n(v)(h) | = d
  2. rewriting for the point |{(P(v)-r(v)} . n(v) |
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10
Q

derivation of a formula for the shortest distance between two lines

A
  1. r(v) = r0(v) + λ a(v)
  2. q(v) = q0(v) + λ b(v)
  3. n(v)(h) = {a(v) x b(v)} / {|a(v) x b(v) |}
  4. d = |(r0(v) - q0(v)) . n(v)(h) |
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11
Q

find an expression for the angular momentum J(v) = r(v) x p(v) rotating with angular velocity ω(v)

A

J(v) = r(v) x p(v) = m r(v) x v(v)

= m r(v) x {ω(v) x r(v)}
= mr^2 ω(v) - m(r(v) . ω(v)) r(v)

=> J(v) = I ω(v)

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12
Q

Differentiating a vector

A
  1. take the derivative of each component
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13
Q

Using the gradient find the location of a scalar functions minimum

A
  1. Take the gradient
  2. each vector component = 0
  3. solve for each component (x,y,z)
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14
Q

the scalar potential φ = x^2 + y^2 + z^2 can be written in cylindrical and polar coordinates as

A

cylindrical = φ = p^2 + z^2

polar = r^2

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15
Q

express the position vector in terms of the unit vectors of cylindrical coordinates

A
  1. equate r(v) = r(v) cylindrical
  2. solve for cylindrical unit vectors
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16
Q

express the cartesian coordinate in terms of the unit vectors of cylindrical coordinates

A
  1. find p = √(x^2+y^2)
  2. find azimuthal angle φ = arctan(y/x)
  3. substitute into r =p e(h)(p) + ze(h)(z)
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17
Q

express the vector field into terms of the unit vectors of cylindrical coordinates

A
  1. compare vector field to cylindrical vector field
  2. vector field = cylindrical vector field
  3. compare cylindrical vector field to r =p e(h)(p) + ze(h)(z)
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18
Q

express the cartesian vector in terms of cylindrical polar variables p, φ and z

A
  1. compare to vector to cylindrical r(v)
  2. r = p e(h)(p) + ze(h)(z)
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19
Q

take a cartesian vector and express it in terms of the unit vectors of cylindrical polar coordinates

A

p = √(x^2+y^2)

φ = arctan(y/x)

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20
Q

Calculate the divergence of the position vector r in cartesian and spherical polar coordinates

A
  1. Cartesian - divergence formula ∇ . r(v)
  2. Spherical r(v) = r e(h)(r)
  3. divergence formula ∇ . r(v)
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21
Q

work out the total surface area of a cylinder of length L and radius R

A
  1. A = curved surface + top + bottom
  2. top and bottom are the same
  3. curved surface + 2top
  4. (L ∫0)(2π ∫0) p dφ dz + 2 (R ∫ 0)(2π ∫ 0) pdpdφ
22
Q

calculate the surface area of a sphere of radius R

A
  1. (2π ∫ 0)(π ∫ 0) R^2 sinθ dθ dφ
23
Q

a string lies along the curve r(v) how long is the length of string between two points

A
  1. take derivative dr(v)/du
  2. ds = |dr(v)/du | du
  3. L = ( B ∫ A) ds
24
Q

evaluate f(x,y) over the path r(v)

A
  1. find the norm ds = |dr(v)/du | du
  2. parameterise f(x,y) using r(x,y)
  3. ( ∫ C) f(x,y) ds
25
for a vector potential evaluate the line integral along the straight line from pont A to B
the integral is (B ∫ A) a(v) . dr(v) where dr(v) = (dx,dy,dz)^T
26
find the flux of a vector field through the surface defined by f(x,y,z)
1. the surface vector dS(v) = (∇ f(x,y,z)) / )∂f(x,y,z)/∂z) dxdy 2. ( ∫ S) a(v) . dS(v)
27
find the volume of an ellipsoid given a surface
1. ellipsoid is a stack of disks of height dz 2. with radius p^2 = x^2 + y^2 3. rewrite volume in correct coords 4. integrate volume over z values
28
demonstrate the divergence theorem
1. first evaluate the LHS of the divergence theorem 2. evaluate the RHS of the divergence theorem 3. if LHS = RHS divergence theorem holds.
29
a unit vector . a unit vector
= 1
30
for a curve and a vector field calculate (∮ C) a(v) . dr(v)
1. stokes theorem (∮ C) a(v) . dr(v) = ( ∫ S) (∇ x a(h)) . dS(v) 2. find the curl of the vector field 3. find normal vector using right hand rule 4. finally solve using stokes theorem ( ∫ S) (∇ x a(h)) . dS(v)
31
consider a function f(x,y) , what is the slope at a point P in x and y direction
1. take the partial derivatives 2. evaluate at P
32
Solve the Laplace equation in 2D using d'Almebert's method
p = x+λt p = λx + y find partial derivatives substitute into original equation solve for λ general form is u(x,y) = u(-) (λ(-)t+x) + u(+)(λ(+)t+x)
33
det of a diagonal matrix is
the product of the diagnals
34
cos(πr)
(-1)^r
35
δ(t-u) =
1/2π (∞ ∫ -∞) exp(iωu)*exp(iωt)
36
scaling property of δ(x)
δ(x) = δ(ay) δ(ay) = 1/a δ(x) [δ(x)] = [x]^-1
37
partial fractions
1. equation = form 2. multiply deniminator 3. solve for roots of form
38
the vector cross-product is
perpendicular to both trajectories so can be regarded as a normal vector
39
distance between two lines
projecting the separating between any point on one line and any other point on the other onto this normal vector d = [r1(v)(0)-r2(v)(0)] . n(hat v)
40
normal vector
n(hat v) = a(v)/|a(v)|
41
directional derivative
∇a(v). n(hat v)
42
positions of sources and sinks
at function = 0
43
solenoidal
div a(v) = 0
44
irrotational
curl a(v) = 0
45
if the curl is = 0
then it is conservative
46
the divergence of the curl
= 0
47
separation of variables
u(x,t) = X(x)T(t) substitute in u(x,t) X(x) on LHS T(t) on RHS general solution
48
∂(x)^2X = -k^2X
X = Acos(kx)+Bsin(kx)
49
∂(x)^2X = k^2X
X = Aexp{kx}+Bexp{-kx}
50
∂(x)X = kX
X = Aexp{kx}
51
T(x,t) =
u(x,t) - u(0)
52