1What is an ordinary differential equation?
A differential equation which contains at least one ordinary derivative
2What is a linear ODE?
An ODE containing only terms which are linear in the dependent: variable (secondary and tertiary derivatives are still linear)
3What is a homogenous linear ODE?
An ODE in which all terms are linearly dependent on the dependent variable
4What is an inhomogoenous linear ODE?
An ODE in which only some terms are linearly dependent on the dependent variable, whilst some terms are independent
5What is the dependent variable?
The function which we are trying to find (i.e. the one which has its derivatives taken)
6What is the independent variable?
The variable which the dependent variable is a function of (i.e. the variable which we take the derivative with respect to)
7What is the general solution?
A way of expressing the solution to the ODE using arbitrary constants so that it represents all possible solutions so that ODE, regardless of initial conditions
8What is a partial differential equation?
A differential equation containing partial derivatives
9What is a linear homogoenous PDE?
A PDE in which each term contains one and only one factor which is a dependent variable
10What is a linear inhomogoenous PDE?
An ODE in which only some terms are linearly dependent on the dependent variable/s, whilst some terms are independent
11What is the wave equation for waves on a line?
$\mathit{\frac{\partial^2\psi}{\partial t^2}-\frac{\partial^2 \psi}{\partial x^2} =0}$
12What is the laplacian?
$\mathit{\Delta\psi(x_1,x_2,x_3) = \frac{\partial ^2\psi}{\partial x_1^2}+\frac{\partial ^2\psi}{\partial x_2^2}+\frac{\partial ^2\psi}{\partial x_3^2}}$
13How may an normalised basis vector be obtained for a generalised coordinate $\mathit{n}$?
$\mathit{\hat{e}_n=\frac{\nabla n}{|\nabla n|}}$
14What is Laplace's equation?
$\mathit{\Delta \psi = 0}$
15What is a spherical harmonic?
The solution to the angular component of the spherical laplacian equation.
16How are spherical harmonics denoted?
$\mathit{Y_n^m(\theta, \phi)}$
17What is the Bessel function of the first kind denoted as?
$\mathit{J_m(x)}$
18What is the Bessel function of the second kind denoted as?
$\mathit{Y_m(x)}$
19What are the properties of the bessel function of the first kind?
$lim_{x\rightarrow 0} \mathit{J_m(x)=J_m(0)}$ is a finite quantity
$lim_{x\rightarrow 0} \mathit{J_m(x)}$ is inifinite
20What are the dimensions of an m x n matrix?
m rows
n columns
21How is the full set of real m x n matrices written?
$\mathit{M_{m\times n}(\mathbb{R})}$
22In index notation, what does $\mathit{M_{ij}$ refer to?
The entry in row i and column j
23What is the norm of a vector?
Its length
24What is the definition of matrix multiplication?
$\mathit{(AB)_{ij}=\sum^n_{k=1} A_{ik}B_{kj}}$
25What is the trace of a matrix?
It is the sum of the diagonal entries of a square matrix
26What is The inverse of the product of two matrices equal to?
The reverse product of the inverses of the two matrices
27What is the transpose?
The mirroring of a matrix along the top-left to bottom-right diagonal
28What is a symmetric matrix?
One which is unafected by the transpose operation
29What is an orthogonal matrix?
One for which the identity $\mathit{A^TA=I_n}$ holds true
30What is a length preserving matrix?
One for which the identity $||\mathit{A\vec{v}}||=||\mathit{\vec{v}}||$ holds true
31What are the requirements for a length preserving matrix?
A matrix is length preserving if and only if if is orthogonal
32What is an orthogonal matrix with a determinant of +1?
A rotation matrix
33What is an orthogonal matrix with a determinant of -1?
A reflection matrix
34What is the hermitian conjugate of a matrix?
Its complex conjugated transpose
35What is a Hermitian matrix?
A matrix which is unchanged by the hermitian conjugate operation
36What is a unitary matrix?
A matrix for which the identity $\mathit{A^\dag A = I}$. It is analagous to rotation and reflection matrices.
37How may be the eigenvalues of a matrix be found?
By solving for the roots of the characteristic polynomial $\mathit{p_A(\lambda)=(-1)^n}\det(\mathit{A-\lambda I)}$
38What is a diagonaliseable matrix?
A matrix which may be expressed as $\mathit{A=B\Lambda B^{-1}}$ where $\mathit{\Lambda}$ is a diagonal matrix made up of the eigenvalues of A, and B is a matrix made up of the column eigenvectors of A
39What is a degenerate or singular matrix?
A matrix with a determinant equal to zero
40What is the relationship between eigenvalue degeneracy and diagonaliseability?
If all eigenvalues of a matrix arre distinct (no repeated roots), then the matrix is diagonaliseable
41What do the eigenvectors of a reflection or scaling matrix represent?
Directions which are unafected (the reflection axis and its normal in the case of a reflection matrix, and the scaling direction and its normal in the case of a scaling matrix)