Raising and lowering indices information

mathematics and mathematical physics, raising and lowering indices are operations on tensors which change their type. Raising and lowering indices are a form...

metric), one can raise and lower indices. A basis gives such a form (via the dual basis), hence when working on Rn with a Euclidean metric and a fixed orthonormal...

(intrinsic definition) Tensor calculus Tensor field While the raising and lowering of indices is dependent on a metric tensor, the covariant derivative is...

simplification at the cost of generality and of some theoretical insight. Contraction of a tensor Raising and lowering indices Symmetric tensor Antisymmetric tensor...

well-behaved and contraction operations make sense in this context. Tensor product Partial trace Interior product Raising and lowering indices Musical isomorphism...

notation Tensor Antisymmetric tensor Raising and lowering indices Covariance and contravariance of vectors Kip S. Thorne and Roger D. Blandford (2017). Modern...

of "lowering the index" on a vector field. The inverse of Sg is a mapping T*M → TM which, analogously, gives an abstract formulation of "raising the index"...

transvectant in invariant theory A shear mapping in linear algebra Raising and lowering indices This disambiguation page lists articles associated with the title...

contravariant; at least one of the indices of a mixed tensor will be a subscript (covariant) and at least one of the indices will be a superscript (contravariant)...

(flat) and ♯ {\displaystyle \sharp } (sharp). In the notation of Ricci calculus, the idea is expressed as the raising and lowering of indices. In certain...

which flips a matrix over its diagonal; that is, it switches the row and column indices of the matrix A by producing another matrix, often denoted by AT (among...

from contravariant entities, see covariance and contravariance of vectors and raising and lowering indices. In several programming languages, index notation...

introductory course), and one needs not be concerned with covariant vectors and contravariant vectors (or raising and lowering indices) to be described below...

particular shape with many lines projecting upwards and downwards, corresponding to abstract upper and lower indices of tensors respectively. Connecting lines between...

covariant components have lower indices, while contravariant components have upper indices. The duality between covariance and contravariance intervenes...

is quite graphically known as lowering an index. Conversely, the inverse operation can be defined, and is called raising an index. This is equivalent to...

used, so repeated indices indicate summation over indices and contraction with the metric tensor serves to raise and lower indices: g ( X , Y ) = X i...

operations of addition, tensor multiplication, tensor contraction, raising and lowering indices, and covariant differentiation may appear in the equation. Forbidden...

weight W . {\displaystyle W.} Using (2) and (3) one sees that raising and lowering indices using the metric tensor (weight 0) leaves the weight unchanged...

takes two equal-length sequences of numbers (usually coordinate vectors), and returns a single number. In Euclidean geometry, the dot product of the Cartesian...

tensor, and vice versa. Example of lowering index using metric tensor: T i = Z i j T j {\displaystyle T_{i}=Z_{ij}T^{j}} Example of raising index using...

theory and supersymmetry. It is named after Bartel Leendert van der Waerden. Undotted indices (chiral indices) Spinors with lower undotted indices have...

number 0, the number of indices is 2, and one of the indices has the value of zero. While the discrete unit sample function and the Kronecker delta function...

extra condition that it is trace-free: metric contraction on any pair of indices yields zero. It is obtained from the Riemann tensor by subtracting a tensor...

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...