Non-Linear Programming: Gradient Descent and Newton’s Method
Non-Linear Programs When visualizing a linear program, its feasible region looks like a polygon. Because the objective function...
Stern-Gerlach Experiments & Dirac Bra-ket Notation
Electron Spin Electron has intrinsic angular momentum called spin that is not associated with its orbital motion. And...
Special Relativity: Space and Time
For hundreds years, space and time were absolute in the concept of Newton, they are something that can...
Quantum Mechanics: 1-Dimensional Finite Potential Problems
Finite Potential Well Finite potential well problem is more realistic compared to the infinite potential well. It is...
Time-Independent Schrödinger Equation
Wave-particle Duality The double-slit experiments show us: particle source P1 : probability pattern when slit 1 is openP2...
Vector Calculus: Fundamental Theorems
Gradient Theorem The fundamental theorem of single variable calculus was the one that told you “the integral of the...
Line and Surface Integrals
Line Integrals Scalar Fields We have a curve C in the x-y plane, we can represent a point...
The Diffusion Equation of a Dye
Fourier Series Sometimes complicated motions can actually be composed of motions of many different frequencies. The type of mathematical analysis...
Systems of Differential Equations
Systems of Homogeneous Linear First-order ODEs The system of linear first order homogeneous equations can be written in matrix...
Polar, Cylindrical, Spherical Coordinates
Multidimensional Integration In Vector Calculus, we have to worry about integrating over two or three variables: double integrals...
Branch-and-Bound & Heuristic Algorithms
In some cases, variables must take integer values, or binary values. Formulating and solving the models with integer...
Laplace Transform and Series Solution Method
Laplace transform Laplace transform is a technique for solving differential equations. By using the Laplace transform you can...
From Partial Derivatives to Maxwell’s Equations
Partial derivatives Partial derivative is to differentiate functions of multiple variables. Assume a function f = f(x, y),...
Logic for Economists
Propositional Logic A proposition (often denoted by capital letters P, Q, etc) is a statement, that can be...
Inhomogeneous Linear Differential Equations
The general form of inhomogenous linear differential equations is: We need 2 initial conditions x(0) = x0, x'(0)...