# 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.
Lesson
Level 3 (with higher mathematics)

## Levi-Civita Symbol and How to Write Cross Product with it

Here you will learn about Levi-Civita symbol; how it is defined and how it can be used to write and prove scalar triple product and cross product.

Video
Level 3 (with higher mathematics)

## Dirac's Delta Function

In this video you will learn what the Dirac delta function is. We also derive its properties and make examples.

Content of the video
1. What is the delta function good for?
2. 02:09 Delta function graphically
3. 02:21 Integral of the delta function at the origin
4. 03:08 Shifted delta function
5. 03:41 Integral of the shifted delta function
6. 04:33 Delta function is symmetric (even)
7. 05:44 Delta function with a factor
8. 06:38 Examples
9. 07:52 Three-dimensional delta function
Formula

## Christoffel Symbols (without Torsion)

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
Video
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
Video
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