Practice Questions

Question 1

How to find the eigenvalues and normalised eigenvectors of a matrix

Answer 1

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

Question 2

Fourier series for a square wave

Answer 2

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

Question 3

Indirect calculation of Laplace transform

Answer 3

1. Apply Laplace transform table

Question 4

Inverse Laplace transform using partial fractions

Answer 4

1. use partial fractions
2. use table to solve

Question 5

solution of ODE with constant coefficients using the Laplace transform

Answer 5

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

Question 6

How to calculate the unit vector

Answer 6

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

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

and a = √(a(x)^2+a(y)^2)

Question 7

how to calculate the projection of b onto the vector a

Answer 7

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

Question 8

what is the normal to the plane

Answer 8

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

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

Question 9

what is the shortest distance from plane

Answer 9

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

Question 10

derivation of a formula for the shortest distance between two lines

Answer 10

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) |

Question 11

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

Answer 11

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)

Question 12

Differentiating a vector

Answer 12

1. take the derivative of each component

Question 13

Using the gradient find the location of a scalar functions minimum

Answer 13

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

Question 14

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

Answer 14

cylindrical = φ = p^2 + z^2

polar = r^2

Question 15

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

Answer 15

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

Question 16

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

Answer 16

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)

Question 17

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

Answer 17

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)

Question 18

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

Answer 18

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

Question 19

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

Answer 19

p = √(x^2+y^2)

φ = arctan(y/x)

Question 20

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

Answer 20

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

Question 21

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

Answer 21

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

Question 22

calculate the surface area of a sphere of radius R

Answer 22

1. (2π ∫ 0)(π ∫ 0) R^2 sinθ dθ dφ

Question 23

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

Answer 23

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

Question 24

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

Answer 24

1. find the norm ds = |dr(v)/du | du
2. parameterise f(x,y) using r(x,y)
3. ( ∫ C) f(x,y) ds

Question 25

for a vector potential evaluate the line integral along the straight line from pont A to B

Answer 25

the integral is (B ∫ A) a(v) . dr(v)

where dr(v) = (dx,dy,dz)^T

Question 26

find the flux of a vector field through the surface defined by f(x,y,z)

Answer 26

1. the surface vector dS(v) = (∇ f(x,y,z)) / )∂f(x,y,z)/∂z) dxdy
2. ( ∫ S) a(v) . dS(v)

Question 27

find the volume of an ellipsoid given a surface

Answer 27

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

Question 28

demonstrate the divergence theorem

Answer 28

1. first evaluate the LHS of the divergence theorem
2. evaluate the RHS of the divergence theorem
3. if LHS = RHS divergence theorem holds.

Question 29

a unit vector . a unit vector

Answer 29

= 1

Question 30

for a curve and a vector field calculate (∮ C) a(v) . dr(v)

Answer 30

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)

Question 31

consider a function f(x,y) , what is the slope at a point P in x and y direction

Answer 31

1. take the partial derivatives
2. evaluate at P

Question 32

Solve the Laplace equation in 2D using d’Almebert’s method

Answer 32

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)

Question 33

det of a diagonal matrix is

Answer 33

the product of the diagnals

Question 34

cos(πr)

Answer 34

(-1)^r

Question 35

δ(t-u) =

Answer 35

1/2π (∞ ∫ -∞) exp(iωu)*exp(iωt)

Question 36

scaling property of δ(x)

Answer 36

δ(x) = δ(ay)
δ(ay) = 1/a δ(x)
[δ(x)] = [x]^-1

Question 37

partial fractions

Answer 37

1. equation = form
2. multiply deniminator
3. solve for roots of form

Question 38

the vector cross-product is

Answer 38

perpendicular to both trajectories so can be regarded as a normal vector

Question 39

distance between two lines

Answer 39

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)

Question 40

normal vector

Answer 40

n(hat v) = a(v)/|a(v)|

Question 41

directional derivative

Answer 41

∇a(v). n(hat v)

Question 42

positions of sources and sinks

Answer 42

at function = 0

Question 43

solenoidal

Answer 43

div a(v) = 0

Question 44

irrotational

Answer 44

curl a(v) = 0

Question 45

if the curl is = 0

Answer 45

then it is conservative

Question 46

the divergence of the curl

Answer 46

= 0

Question 47

separation of variables

Answer 47

u(x,t) = X(x)T(t)

substitute in u(x,t)

X(x) on LHS T(t) on RHS

general solution

Question 48

∂(x)^2X = -k^2X

Answer 48

X = Acos(kx)+Bsin(kx)

Question 49

∂(x)^2X = k^2X

Answer 49

X = Aexp{kx}+Bexp{-kx}

Question 50

∂(x)X = kX

Answer 50

X = Aexp{kx}

Question 51

T(x,t) =

Answer 51

u(x,t) - u(0)