Lecture

Tensors & Indices Recap

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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 basics of tensors and indices, including contravariant vectors, Minkowski metric, covariant vectors, and tensor operations. The instructor explains key concepts with examples and 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

Mathematics

Analysis: Tensor analysis

Related concepts (29)

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.

Four-vector

In special relativity, a four-vector (or 4-vector) is an object with four components, which transform in a specific way under Lorentz transformations. Specifically, a four-vector is an element of a four-dimensional vector space considered as a representation space of the standard representation of the Lorentz group, the (1/2,1/2) representation. It differs from a Euclidean vector in how its magnitude is determined.

Euclidean vector

In mathematics, physics, and engineering, a Euclidean vector or simply a vector (sometimes called a geometric vector or spatial vector) is a geometric object that has magnitude (or length) and direction. Vectors can be added to other vectors according to vector algebra. A Euclidean vector is frequently represented by a directed line segment, or graphically as an arrow connecting an initial point A with a terminal point B, and denoted by . A vector is what is needed to "carry" the point A to the point B; the Latin word vector means "carrier".

Tensor (intrinsic definition)

In mathematics, the modern component-free approach to the theory of a tensor views a tensor as an abstract object, expressing some definite type of multilinear concept. Their properties can be derived from their definitions, as linear maps or more generally; and the rules for manipulations of tensors arise as an extension of linear algebra to multilinear algebra. In differential geometry an intrinsic geometric statement may be described by a tensor field on a manifold, and then doesn't need to make reference to coordinates at all.

Exterior covariant derivative

In the mathematical field of differential geometry, the exterior covariant derivative is an extension of the notion of exterior derivative to the setting of a differentiable principal bundle or vector bundle with a connection. Let G be a Lie group and P → M be a principal G-bundle on a smooth manifold M. Suppose there is a connection on P; this yields a natural direct sum decomposition of each tangent space into the horizontal and vertical subspaces. Let be the projection to the horizontal subspace.

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