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In general, indices can range over any indexing set, including an infinite set. This should not be confused with a typographically similar convention used to distinguish between tensor index notation and the closely related but distinct basis-independent abstract index notation. An index that is summed over is a summation index, in this case "i ".
It is common convention to use greek indices when writing expressions involving tensors in Minkowski space, while Latin indices are reserved for Euclidean space. Well-formulated expressions are constrained by the rules of Einstein summation : any index may appear at most twice and furthermore a raised index must contract with a lowered index.
In multilinear algebra, a tensor contraction is an operation on a tensor that arises from the canonical pairing of a vector space and its dual.In components, it is expressed as a sum of products of scalar components of the tensor(s) caused by applying the summation convention to a pair of dummy indices that are bound to each other in an expression.
The entry of a matrix A is written using two indices, say i and j, with or without commas to separate the indices: a ij or a i,j, where the first subscript is the row number and the second is the column number. Juxtaposition is also used as notation for multiplication; this may be a source of confusion. For example, if
Multi-index notation is a mathematical notation that simplifies formulas used in multivariable calculus, partial differential equations and the theory of ...
The pocket-sized Hewlett-Packard HP-35 scientific calculator was the first handheld device of its type, but it cost US$395 in 1972. This was justifiable for some engineering professionals, but too expensive for most students. Around 1974, lower-cost handheld electronic scientific calculators started to make slide rules largely obsolete.
A mixed tensor of type or valence (), also written "type (M, N)", with both M > 0 and N > 0, is a tensor which has M contravariant indices and N covariant indices. Such a tensor can be defined as a linear function which maps an ( M + N )-tuple of M one-forms and N vectors to a scalar .
Matrix multiplication shares some properties with usual multiplication. However, matrix multiplication is not defined if the number of columns of the first factor differs from the number of rows of the second factor, and it is non-commutative, [10] even when the product remains defined after changing the order of the factors. [11] [12]