# Mathematics

Learn the arts of numbers, logical definitions, and constructed abstract structures, as well as their relationships to each other, which will help you better master physics.
Video
Level 3 (with higher mathematics)

## Hermitian Operators Simply Explained

In this physics video you will learn what Hermitian operators and matrices are and what important properties they have.

Content of the video
1. 00:00 Motivation: Real mean values
2. 01:16 What are Hermitian operators?
3. 02:16 What is a self-adjoint operator?
4. 02:56 Notation of a Hermitian operator
5. 03:20 Eigenvalues are real
6. 04:27 Eigenvectors are orthogonal
7. 05:56 Eigenvectors form a basis
8. 06:37 What is a Hermite matrix?
9. 07:19 Examples for (non) Hermitian matrices
Lesson
Level 3 (with higher mathematics)

## Separation of Variables (SoV) and How to Solve Homogeneous DEQ's of 1st Order

Learn how to solve first order homogeneous differential equations using the separation of variables (SoV) method. With example for the decay law.

Lesson
Level 3 (with higher mathematics)

## Variation of Constants and How to Solve Inhomogeneous Differential Equations of 1st Order

Learn "Variation of constants" - solution method. With its solution formula you can solve ordinary inhomogeneous differential equations of 1st order.

Lesson
Level 3 (with higher mathematics)

## Exponential Ansatz and How to Solve Linear Differential Equations (of 2nd order)

Learn how to solve linear differential equations (2nd order) using the exponential approach and how to use the characteristic equation (+ example).

Video
Level 3 (with higher mathematics)

## How to Solve Linear 2nd Order Differential Equation with Characteristic Polynomial ( + Example)

Learn how to solve 2nd order (in)homogeneous linear differential equations using the exponential ansatz. The DEQ of the harmonic oscillator is solved.

Content of the video
1. [00:00] What is exponential ansatz good for?
2. [00:45] Derivation of the method
3. [03:09] Example: Undamped harmonic oscillator
Lesson
Level 3 (with higher mathematics)

## Separation of Variables and How to Solve Partial Differential Equations

Separation of variables (product approach) is suitable for converting partial differential equations (PDE's) into ordinary differential equations and then solving them with other methods.

Video
Level 3 (with higher mathematics)

## How to Solve Partial Differential Equations with Separation of Variables

Learn how to solve partial differential equations (PDE) with separation of variables (product ansatz). Here we make an example with the one-dimensional wave equation.

Content of the video
1. [00:00] What is Separation of Variables good for?
2. [00:23] Example: Separate 1d wave equation
Level 3 (with higher mathematics)

## How to Solve 1st Order Inhomogeneous Differential Equations (with Variation of Constants)

Learn how to solve 1st order inhomogeneous linear differential equations using the "variation of constants" (variation of parameters) method.

Content of the video
1. [00:00] What is Variation of Parameters good for?
2. [00:45] Derivation of the method
3. [03:09] Example: RL circuit
Level 3 (with higher mathematics)

## How to Solve 1st Order Homogeneous Differential Equations (with Separation of Variables)

Here you will learn how to solve 1st order homogeneous linear differential equations using the "separation of variables" solving method.

Content of the video
1. [00:00] What is Separation of Variables good for?
2. [00:39] Derivation of the method
3. [02:25] Example: Decay law
Video
Level 3 (with higher mathematics)

## Divergence Theorem: The Simplest Explanation Every Physicist Should Know

Learn the Divergence Theorem in only 8 minutes, namely how it links a volume integral with a surface integral.

Content of the video
1. [00:00] Equation
2. [00:05] Surface integral in the Divergence Theorem
3. [04:21] Volume integral in the Divergence Theorem
4. [07:09] Summary
Video
Level 3 (with higher mathematics)

## Differential Equations explained simply in 47 minutes!

In this Lecture you will learn to classify and to solve arbitrary differential equations (ODE / PDE) in 42 minutes.

Content of the video
1. [00:16] Why do I need differential equations?
2. [00:47​] What is a differential equation?
3. [03:41​] Different notations of a differential equation
4. [05:19​] What should I do with a differential equation?
5. [06:31​] How to identify a differential equation
6. [07:28​] What are coupled differential equations?
7. [08:53​] Classification: Which DEQ types are there?
8. [16:05​] What are DEQ constraints?
9. [18:24] Difference between boundary and initial conditions
10. [20:55​] Solving method #1: Separation of variables
11. [21:08​] Example: Radioactive Decay law
12. [22:50​] Solving method #2: Variation of constants
13. [25:59​] Example: RL Circuit
14. [29:14​] Solving method #3: Exponential ansatz
15. [34:50​] Example: Oscillating Spring
16. [41:15​] Solving method #4: Product / Separation ansatz
Lesson
Level 3 (with higher mathematics)

## Differential Equations (DEQ): All the Basics Everyone Should Know

Learn what differential equations are, what types there are (ordinary, partial, linear, homogeneous), and what boundary and initial conditions are good for.

Course
Level 2 (without higher mathematics)

## Mathematics for Physics Enthusiasts I

Free online course where you learn basic math that a physicist should know in order to do physics at all.