## Kronecker Delta: 4 Important Rules and Scalar Product in Index Notation

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

Advanced tools from mathematics for physicists.

Level 3 requires the basics of vector calculus, differential and integral calculus. Suitable for undergraduates and high school students.

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Lesson ## Kronecker Delta: 4 Important Rules and Scalar Product in Index Notation

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

## Quests with solutions

Exercise with solution ## Simplify 6 Expressions with Kronecker Delta

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Lesson ## 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.

## Related Illustrations

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Lesson ## Dirac's Delta Function and its Most Important Properties

Here you will learn about the delta function and its properties, which you can use for example to describe an electric point charge.

## Quests with solutions

Exercise with solution ## Simplify 6 Integrals with the Delta Function

## Related Illustrations

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Lesson ## Fourier Series (+ Coefficients) and How to Intuitively Understand it

Here you will learn what a Fourier series is, how it can be calculated, and the role of Fourier coefficients and basis functions.

## Related Illustrations

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Lesson ## 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.

## Related Illustrations

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Lesson ## 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.

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Lesson ## 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.

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Lesson ## 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).

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Lesson ## 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.

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Lesson ## Nabla Operator and its 3 Most Important Applications

Learn Nabla operator to represent and calculate gradient (grad), divergence (div), rotation (curl) and other operators.

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Lesson ## Gradient and How to Calculate the Directional Derivative

Learn to calculate the gradient of a function using the Nabla operator and to determine the directional derivative.

## Related formulas

Formula ## Gradient of a scalar function

`$$ \nabla \, f(x,y,z) ~=~ \begin{bmatrix} \frac{\partial f}{\partial x} \\ \frac{\partial f}{\partial y} \\ \frac{\partial f}{\partial z} \end{bmatrix} $$`## Related Illustrations

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Mathematical proof ## First and Second Green's Identity

Proof of the two Green's identities (formulas) using the Gauss integral theorem, which are useful in the calculation of some electric potentials.

## Derivations & Experiments

Mathematical proof ## First and Second Green's Identity

Proof of the two Green's identities (formulas) using the Gauss integral theorem, which are useful in the calculation of some electric potentials.

## Related Illustrations