Some Git memos

A tool to display and compare conflicting Jupyter files in different commits: nbdime Example usage (assuming at DES branch and want to compare it to origin/main):

CS 124 Java functions

This page contains all the objects, functions, methods, and their required input types that are included in the CS 124 course. syntax basic objects and methods Class and Interface copied from

Electromagnetics elementary formulae

Mathematical technique Vector algebra dot product: Cross product: Differential Calculus The gradient: The position vector : Distance between 2 points and : The gradient on : The divergence: The curl: Product rules: 2 for gradients/curls/divergences: 2nd divergences (def of vector Laplacian): Suffix notation Tensor (notations in this document): In the operation of tensors, it is […]

Frobenius method

ODE Frobenius method Consider an ODE has the form The solution can be represented in the form Bessel’s Equation: Regular point and singular point – Indicial equation Rearrange the equation above If and are analytic, then it can be expanded as power series Let Insert this into (*) At the term, we have Assuming , […]

Integral Transforms

Fourier series & Transform Example: for , in Then, Parseval’s theorem: Fourier transform f(x) satisfied dirichlet Condition f(x) must be periodic, with fundamental period Q: Can we find F.S, for with no periodic? A: Yes! F.T. which is one of the integral transform Arrangement: introduction of F.T. Laplace transform (other integral transform Suppose that is […]

Fourier Series

idea: to express complicated functions by using simple functions for example: power series : Besides power series, we do have other ways, such as Fourier series Advantages: easy to differentiate and integrate each term contain only one characteristic frequency To solve PDE and ODE (forbid some numerical methods) can be used to describe non continuous […]

Line, surface and volume integrals

Line integral Recall: Riemann integral Notation: Def: Line integral: We denote path joining A and B into N small line element , If a field is any point on the line element , then Remark: If is closed, then where is anticlockwise Properties: (could be) path independent How to evaluate: Example: evaluate , where , […]

Vector Calculus

Differentiation of vectors Def: vector that is a function of a scalar , Differentiation of compose vector expression Let be vector function of , be scalar function of Then, Differentiation of a vector We use a small change in a vector function resulting from a small change in its argument. When , For example, for […]

Matrices and vector spaces

Review: Vector algebra Def: Hilbert Space Def: Linear dependant: Suppose vectors: If NOT ALL ZEROS , that Linear independant: Inner Product   Properties: Def: norm: Orthogonal: Kronecker delta symbol: some useful inequalities Schwarz’s inequality: where the equality holds when a is a scalar multiple of b, i.e. when . Proof: (p. 246) Hint: suqare the […]