Illustration Gradient of a Linear Scalar Function (Homogeneous Vector Field)

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This is how the vector field $\nabla \varphi $ (gradient) looks like in the range between $x,y \in [0,1]$, where the scalar function $\varphi$ is defined as follows:$$ \varphi(x,y) = x $$

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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

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

CourseMathematics for physics enthusiasts II Advanced tools from mathematics for physicists.

All contents of the course

Lesson
Level 3 (with higher mathematics)
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.
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.
Lesson
Level 3 (with higher mathematics)
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.
Lesson
Level 3 (with higher mathematics)
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.
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.
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).
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.
Lesson
Level 3 (with higher mathematics)
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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Illustration Gradient of a Linear Scalar Function (Homogeneous Vector Field)

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

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

Dirac's Delta Function and its Most Important Properties

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

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

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

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

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

Separation of Variables and How to Solve Partial Differential Equations

Nabla Operator and its 3 Most Important Applications