![]() ![]() The next article in this series is Tensors/Bases, components, and dual spaces.See the guide for this topic. The principle of gravitation in General relativity simply says that Einstein's tensor is 8πK times the stress-energy-momentum tensor, where K is Newton's constant of gravitation. Ricci's tensor and Einstein's tensor, 2 nd rank covariant symmetric, are simplified versions of Riemann's tensor, describe the curvature of spacetime, and make General relativity work. The stress-energy-momentum tensor 2 nd rank covariant symmetric, is the tensor in 4-dimensional relativistic spacetime that describes all the stresses, forces, momenta, matter, and energy. Riemann's tensor, 4 th rank mixed, is made from the derivatives (gradients) of the metric tensor in different parts of space (that is, a tensor field), and describes the curvature of the space. ![]() The Electromagnetic stress-energy tensor generalizes this to 4 dimensions in relativity and describes the energy and momentum densities of the electromagnetic field. ![]() The Maxwell stress tensor, 2 nd rank contravariant symmetric, is the tensor in 3 dimensions that describes the classical stress of the electric and magnetic fields. The stress tensor, 2 nd rank covariant symmetric, is the tensor in 3 dimensions that describes the mechanical stresses on an object. Its components are the components of the classical electric and magnetic fields. As such, it is a pseudo-tensor.įaraday's tensor, 2 nd rank contravariant antisymmetric, is the tensor that explains electrodynamics and Maxwell's Equations in 4-dimensional relativistic spacetime. It has the unusual property of having its sign depend on the "handedness" of the space. ![]() The Levi-Civita tensor, N th rank covariant, completely antisymmetric (N is the dimension of the space), measures the area/volume/hypervolume of the rectangle/parallelogram/parallelopiped/parallelotope spanned by N vectors. Being a symmetric bilinear function of two vectors, it is just the right thing for defining a dot product. The metric tensor, or just the metric, 2 nd rank covariant symmetric, is very commonly used to define the "inner product" or " dot product" of two vectors. Symmetry and antisymmetry are only meaningful if the arguments being swapped are of the same type-both covariant or both contravariant. This is similar to the anticommutative property of subtraction. A tensor is antisymmetric if it gives the negative of the result when the two arguments are switched. This is similar to the commutative property of addition. Since Q takes a form as its argument, it is a 1 st rank contravariant tensor, which is a vector, as shown in the next article.Ī tensor is symmetric in a given pair of arguments if it gives the same result when those two arguments are switched with each other. These terms need to be explained carefully. A tensor is a real-valued function of some number of vectors and/or linear forms, which is linear in each of its arguments. ![]()
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