PHYS211 - Maths I

A differential equation which contains at least one ordinary derivative

An ODE containing only terms which are linear in the dependent: variable (secondary and tertiary derivatives are still linear)

An ODE in which all terms are linearly dependent on the dependent variable

An ODE in which only some terms are linearly dependent on the dependent variable, whilst some terms are independent

The function which we are trying to find (i.e. the one which has its derivatives taken)

The variable which the dependent variable is a function of (i.e. the variable which we take the derivative with respect to)

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

A differential equation containing partial derivatives

A PDE in which each term contains one and only one factor which is a dependent variable

An ODE in which only some terms are linearly dependent on the dependent variable/s, whilst some terms are independent

$\mathit{\frac{\partial^2\psi}{\partial t^2}-\frac{\partial^2 \psi}{\partial x^2} =0}$

$\mathit{\Delta\psi(x_1,x_2,x_3) = \frac{\partial ^2\psi}{\partial x_1^2}+\frac{\partial ^2\psi}{\partial x_2^2}+\frac{\partial ^2\psi}{\partial x_3^2}}$

$\mathit{\hat{e}_n=\frac{\nabla n}{|\nabla n|}}$

$\mathit{\Delta \psi = 0}$

The solution to the angular component of the spherical laplacian equation.

$\mathit{Y_n^m(\theta, \phi)}$

$\mathit{J_m(x)}$

$\mathit{Y_m(x)}$

$lim_{x\rightarrow 0} \mathit{J_m(x)=J_m(0)}$ is a finite quantity
$lim_{x\rightarrow 0} \mathit{J_m(x)}$ is inifinite

m rows
n columns

$\mathit{M_{m\times n}(\mathbb{R})}$

The entry in row i and column j

Its length

$\mathit{(AB)_{ij}=\sum^n_{k=1} A_{ik}B_{kj}}$

It is the sum of the diagonal entries of a square matrix

The reverse product of the inverses of the two matrices

The mirroring of a matrix along the top-left to bottom-right diagonal

One which is unafected by the transpose operation

One for which the identity $\mathit{A^TA=I_n}$ holds true

One for which the identity $||\mathit{A\vec{v}}||=||\mathit{\vec{v}}||$ holds true

A matrix is length preserving if and only if if is orthogonal

A rotation matrix

A reflection matrix

Its complex conjugated transpose

A matrix which is unchanged by the hermitian conjugate operation

A matrix for which the identity $\mathit{A^\dag A = I}$. It is analagous to rotation and reflection matrices.

By solving for the roots of the characteristic polynomial $\mathit{p_A(\lambda)=(-1)^n}\det(\mathit{A-\lambda I)}$

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

A matrix with a determinant equal to zero

If all eigenvalues of a matrix arre distinct (no repeated roots), then the matrix is diagonaliseable

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)