Practice Questions Flashcards
How to find the eigenvalues and normalised eigenvectors of a matrix
- Solve the characteristic equation for A
- Solutions are the eigenvalues
- The eigenvector corresponding to the eigenvalue has to satisfy (A-λI)x(v) = 0
(v) is the vector
Fourier series for a square wave
- Fourier series expansion formula
- Find the coefficients
- odd -> cos is odd -> an = 0
- even -> sin is even -> bn = 0
Indirect calculation of Laplace transform
- Apply Laplace transform table
Inverse Laplace transform using partial fractions
- use partial fractions
- use table to solve
solution of ODE with constant coefficients using the Laplace transform
- Take the Laplace transform
- Substitute in the boundary conditions and rearrange
- Use partial fractions
- use table to solve
How to calculate the unit vector
a(v)(h) = a(v)/a
where a(v)(h) is the unit vector
and a = √(a(x)^2+a(y)^2)
how to calculate the projection of b onto the vector a
- the component of b that is in the direction of a
- evaluate the dot product a(hat) . b
what is the normal to the plane
- solve | r(v) . n(v)(h) | = d
where n(v) is the coefficients of the equation.
and r(v) = (x,y,z)
what is the shortest distance from plane
- | r(v) . n(v)(h) | = d
- rewriting for the point |{(P(v)-r(v)} . n(v) |
derivation of a formula for the shortest distance between two lines
- r(v) = r0(v) + λ a(v)
- q(v) = q0(v) + λ b(v)
- n(v)(h) = {a(v) x b(v)} / {|a(v) x b(v) |}
- d = |(r0(v) - q0(v)) . n(v)(h) |
find an expression for the angular momentum J(v) = r(v) x p(v) rotating with angular velocity ω(v)
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)
Differentiating a vector
- take the derivative of each component
Using the gradient find the location of a scalar functions minimum
- Take the gradient
- each vector component = 0
- solve for each component (x,y,z)
the scalar potential φ = x^2 + y^2 + z^2 can be written in cylindrical and polar coordinates as
cylindrical = φ = p^2 + z^2
polar = r^2
express the position vector in terms of the unit vectors of cylindrical coordinates
- equate r(v) = r(v) cylindrical
- solve for cylindrical unit vectors
express the cartesian coordinate in terms of the unit vectors of cylindrical coordinates
- find p = √(x^2+y^2)
- find azimuthal angle φ = arctan(y/x)
- substitute into r =p e(h)(p) + ze(h)(z)
express the vector field into terms of the unit vectors of cylindrical coordinates
- compare vector field to cylindrical vector field
- vector field = cylindrical vector field
- compare cylindrical vector field to r =p e(h)(p) + ze(h)(z)
express the cartesian vector in terms of cylindrical polar variables p, φ and z
- compare to vector to cylindrical r(v)
- r = p e(h)(p) + ze(h)(z)
take a cartesian vector and express it in terms of the unit vectors of cylindrical polar coordinates
p = √(x^2+y^2)
φ = arctan(y/x)
Calculate the divergence of the position vector r in cartesian and spherical polar coordinates
- Cartesian - divergence formula ∇ . r(v)
- Spherical r(v) = r e(h)(r)
- divergence formula ∇ . r(v)
work out the total surface area of a cylinder of length L and radius R
- A = curved surface + top + bottom
- top and bottom are the same
- curved surface + 2top
- (L ∫0)(2π ∫0) p dφ dz + 2 (R ∫ 0)(2π ∫ 0) pdpdφ
calculate the surface area of a sphere of radius R
- (2π ∫ 0)(π ∫ 0) R^2 sinθ dθ dφ
a string lies along the curve r(v) how long is the length of string between two points
- take derivative dr(v)/du
- ds = |dr(v)/du | du
- L = ( B ∫ A) ds
evaluate f(x,y) over the path r(v)
- find the norm ds = |dr(v)/du | du
- parameterise f(x,y) using r(x,y)
- ( ∫ C) f(x,y) ds
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
find the flux of a vector field through the surface defined by f(x,y,z)
- the surface vector dS(v) = (∇ f(x,y,z)) / )∂f(x,y,z)/∂z) dxdy
- ( ∫ S) a(v) . dS(v)
find the volume of an ellipsoid given a surface
- ellipsoid is a stack of disks of height dz
- with radius p^2 = x^2 + y^2
- rewrite volume in correct coords
- integrate volume over z values
demonstrate the divergence theorem
- first evaluate the LHS of the divergence theorem
- evaluate the RHS of the divergence theorem
- if LHS = RHS divergence theorem holds.
a unit vector . a unit vector
= 1
for a curve and a vector field calculate (∮ C) a(v) . dr(v)
- stokes theorem (∮ C) a(v) . dr(v) = ( ∫ S) (∇ x a(h)) . dS(v)
- find the curl of the vector field
- find normal vector using right hand rule
- finally solve using stokes theorem ( ∫ S) (∇ x a(h)) . dS(v)
consider a function f(x,y) , what is the slope at a point P in x and y direction
- take the partial derivatives
- evaluate at P
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)
det of a diagonal matrix is
the product of the diagnals
cos(πr)
(-1)^r
δ(t-u) =
1/2π (∞ ∫ -∞) exp(iωu)*exp(iωt)
scaling property of δ(x)
δ(x) = δ(ay)
δ(ay) = 1/a δ(x)
[δ(x)] = [x]^-1
partial fractions
- equation = form
- multiply deniminator
- solve for roots of form
the vector cross-product is
perpendicular to both trajectories so can be regarded as a normal vector
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)
normal vector
n(hat v) = a(v)/|a(v)|
directional derivative
∇a(v). n(hat v)
positions of sources and sinks
at function = 0
solenoidal
div a(v) = 0
irrotational
curl a(v) = 0
if the curl is = 0
then it is conservative
the divergence of the curl
= 0
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
∂(x)^2X = -k^2X
X = Acos(kx)+Bsin(kx)
∂(x)^2X = k^2X
X = Aexp{kx}+Bexp{-kx}
∂(x)X = kX
X = Aexp{kx}
T(x,t) =
u(x,t) - u(0)
