Lecture

Covariant and Contravariant Tensors: Maxwell Field Strength

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In course

PHYS-324: Classical electrodynamics

The goal of this course is the study of the physical and conceptual consequences of Maxwell equations. ...

Description

This lecture covers the concepts of contravariant and covariant tensors, with examples of rotational and Lorentz invariance. It delves into the Maxwell tensor or field strength and addresses questions from students regarding linear transformations, metric properties, and covariant Maxwell equations. The lecture also explores the properties of the Maxwell tensor, the construction of invariants, and the manifestation of Lorentz invariance in various equations.

Instructor

Victor Gorbenko

Teaching & PhD PhD Students Jonas Dujava, Anders Wallberg, Mattia Varrone, Aleksandr Zhabin, Veronica Sacchi Courses Classical electrodynamics PHYS-324 The goal of this course is the study of the physical and conceptual consequences of Maxwell equations. Relativity and cosmology II PHYS-428 This course is the basic introduction to modern cosmology. It introduces students to the main concepts and formalism of cosmology, the observational status of Hot Big Bang theory and dis

Official source

Ontological neighbourhood

Physics

Electromagnetism: Electrodynamics

Related concepts (32)

Covariance and contravariance of vectors

In physics, especially in multilinear algebra and tensor analysis, covariance and contravariance describe how the quantitative description of certain geometric or physical entities changes with a change of basis. In modern mathematical notation, the role is sometimes swapped. A simple illustrative case is that of a vector. For a vector, once a set of basis vectors has been defined, then the components of that vector will always vary opposite to that of the basis vectors. A vector is therefore a contravariant tensor.

Covariant derivative

In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. Alternatively, the covariant derivative is a way of introducing and working with a connection on a manifold by means of a differential operator, to be contrasted with the approach given by a principal connection on the frame bundle – see affine connection. In the special case of a manifold isometrically embedded into a higher-dimensional Euclidean space, the covariant derivative can be viewed as the orthogonal projection of the Euclidean directional derivative onto the manifold's tangent space.

Maxwell's equations

Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits. The equations provide a mathematical model for electric, optical, and radio technologies, such as power generation, electric motors, wireless communication, lenses, radar, etc. They describe how electric and magnetic fields are generated by charges, currents, and changes of the fields.

Tensor

In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tensors. There are many types of tensors, including scalars and vectors (which are the simplest tensors), dual vectors, multilinear maps between vector spaces, and even some operations such as the dot product.

Antisymmetric tensor

In mathematics and theoretical physics, a tensor is antisymmetric on (or with respect to) an index subset if it alternates sign (+/−) when any two indices of the subset are interchanged. The index subset must generally either be all covariant or all contravariant. For example, holds when the tensor is antisymmetric with respect to its first three indices. If a tensor changes sign under exchange of each pair of its indices, then the tensor is completely (or totally) antisymmetric.

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