Mathematics

Course Level 3
Eine zweidimensionale Skalarfunktion - 3D Plot

Here you will learn the basics of multidimensional differentiation and integration as well as how to deal with differential equations. These are important tools in physics.

Lesson Level 3

Differential equations (DEQ): Important basics + 4 solving methods

Differentialgleichungen - Klassifizierung

Here you will learn the basics of differential equations (DGL), what types there are (ordinary, partial, linear, homogeneous) and 4 solving methods (separation of variables, variation of constants, exponential and separation ansatz).

Lesson Level 3

Kronecker delta: How to write scalar product in index notation

Orthonormale Basisvektoren - kartesisches Koordinatensystem

Here you learn everything about Kronecker-Delta! Including 4 calculation rules with Einstein's summation convention, typical mistakes and more.

Videos

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Video Level 3

Differential Equations explained simply in 47 minutes!

Differential equations

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
Video Level 3

Kronecker delta explained simply in 11 minutes!

Kronecker delta

In this video you will learn the Kronecker delta within 10 minutes and how you can use it to write down the scalar product in index notation.

Content of the video
  1. [00:00] Hello there!
  2. [00:42] Definition and Examples
  3. [01:52] Einstein's Summation convention
  4. [02:49] 4 Rules for Kronecker delta
  5. [06:38] Scalar product with Kronecker delta

Quests

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Illustrations

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Total set and empty set as elements of a topology
Elements of a topology
Total set and empty set as elements of a topology
Elements of a topology
Total set and empty set as elements of a topology
Elements of a topology
Difference between initial and boundary conditions
Difference between initial conditions and boundary conditions
Differential equations - different types
Classification of differential equations
Orthonormal basis vectors - Cartesian coordinate system
Orthonormal basis vectors - Cartesian coordinate system
Levi-Civita symbol: Cyclic (even) permutation of indices
Levi-Civita symbol: Cyclic (even) permutation of indices
Vectors and vector components - orthogonal coordinate system
Vectors and vector components - orthogonal coordinate system
Graph of a shifted Dirac delta function (1d)
Graph of a shifted Dirac delta function (1d)
Closed line integral of a vector field
Closed line integral of a vector field
Curl Integral Theorem
Curl Integral Theorem
Surface integral with orthogonal surface vector + vector field
Surface integral with orthogonal surface vector + vector field
Divergence theorem
Divergence Integral Theorem (Divergence + Flux)