"A tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space."
The study of mathematical objects that can represent physical quantities, including scalars, vectors, and tensors.
Linear Algebra: Linear algebra is a branch of mathematics that deals with vector spaces and linear transformations. It is the foundation of tensor analysis.
Multivariable Calculus: Multivariable calculus is the extension of calculus in multiple dimensions. It deals with the topics like partial derivatives, gradient, divergence, and curl.
Tensor Algebra: It is a branch of mathematics that describes tensors and their properties. It deals with the concept of linear transformations and higher-order tensors.
Tensor Calculus: Tensor calculus is an extension of vector calculus. It is used to describe the rules for manipulating tensors, including differentiation and integration.
Riemannian Geometry: Riemannian geometry is the study of curves and surfaces in multi-dimensional spaces, including the study of the curvature of surfaces.
Differential Geometry: Differential geometry is the study of curves and surfaces in multi-dimensional spaces, including the study of curvature, geodesics, and tensor fields.
Lie Groups: Lie groups are groups that are also smooth manifolds, with a compatible differentiable structure.
Lie Algebras: Lie algebras are vector spaces equipped with a binary operation called the Lie bracket.
Differential Equations: Differential equations are equations that involve derivatives of a function or functions. It includes partial differential equations, ordinary differential equations, and partial differential equations.
Group Theory: Group theory is the study of symmetry, including the study of abstract groups, group representations, and Lie groups.
Topology: Topology is the study of the properties of space that are preserved under continuous transformations, including the study of topological spaces and manifolds.
Functional Analysis: Functional analysis is the study of infinite-dimensional vector spaces and their properties, including the study of Banach and Hilbert spaces and their operators.
Category Theory: Category theory is the study of structures, including the study of universal properties, functors, and natural transformations.
Nonlinear Optics: Nonlinear optics is the study of the interaction of light with matter, including the study of lasers, nonlinear materials, and nonlinear optical phenomena.
Quantum Mechanics: Quantum mechanics is the study of the behavior of particles at the atomic and subatomic level, including the study of wave-particle duality, quantum entanglement, and quantum computing.
Tensor Calculus: Used for the manipulation of tensors in multi-dimensional Euclidean space.
Riemannian Geometry: The study of geometry based on Riemannian manifolds, in which tensors play an essential role.
Differential Geometry: Deals with the geometry of smooth curves and surfaces, involving a variety of tensor formalisms.
Topology: The study of shape and the properties of space, employing tensors as mathematical objects.
Algebraic Topology: Deals with the use of algebraic invariants to study topological spaces.
Homological Algebra: The study of algebraic structures through homology computations, employing tensors and the cup product.
Category Theory: A branch of pure mathematics that deals with abstract mathematical structures, with importance placed on the interactions of functors, categories and natural transformations using tensor products.
Quantum Mechanics: The branch of physics that explores the behavior of matter at microscopic levels, using tensor analysis to describe the state of a quantum system.
General Relativity: The theory of gravity, where tensors play a central role in describing the curvature of space-time.
Nonlinear Partial Differential Equations: Mathematical modeling of complex systems employing tensor calculus.
"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."
"Tensors are defined independent of any basis, although they are often referred to by their components in a basis related to a particular coordinate system."
"Those components form an array, which can be thought of as a high-dimensional matrix."
"Tensors have become important in physics because they provide a concise mathematical framework for formulating and solving physics problems in areas such as mechanics, electrodynamics, general relativity, and others."
"In applications, it is common to study situations in which a different tensor can occur at each point of an object; for example, the stress within an object may vary from one location to another."
"This leads to the concept of a tensor field. In some areas, tensor fields are so ubiquitous that they are often simply called 'tensors'."
"Tullio Levi-Civita and Gregorio Ricci-Curbastro popularised tensors in 1900."
"The concept enabled an alternative formulation of the intrinsic differential geometry of a manifold in the form of the Riemann curvature tensor."
"[They continued] the earlier work of Bernhard Riemann, Elwin Bruno Christoffel, and others."
"Tensors were popularized [...] as part of the absolute differential calculus."
"[Tensors are used] in areas such as mechanics (stress, elasticity, fluid mechanics, moment of inertia, ...)."
"[Tensors are used] in areas such as electrodynamics (electromagnetic tensor, Maxwell tensor, permittivity, magnetic susceptibility, ...)."
"[Tensors are used] in areas such as general relativity (stress–energy tensor, curvature tensor, ...)."
"The stress within an object may vary from one location to another."
"In some areas, tensor fields are so ubiquitous that they are often simply called 'tensors'."
"Tensors provide a concise mathematical framework for formulating and solving physics problems."
"Tensors are defined independent of any basis, although they are often referred to by their components in a basis related to a particular coordinate system."
"Those components form an array, which can be thought of as a high-dimensional matrix."