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A vector treated as an array of numbers by writing as a row vector or column vector (whichever is used depends on convenience or context): = (), = Index notation allows indication of the elements of the array by simply writing a i, where the index i is known to run from 1 to n, because of n-dimensions. [1]
Concretely, in the case where the vector space has an inner product, in matrix notation these can be thought of as row vectors, which give a number when applied to column vectors. We denote this by V ∗ := Hom ( V , K ) {\displaystyle V^{*}:={\text{Hom}}(V,K)} , so that α ∈ V ∗ {\displaystyle \alpha \in V^{*}} is a linear map α : V → K ...
The above result can be written in index notation as follows. The elements of the matrix for an active rotation by an angle θ {\displaystyle \theta } about an axis n are given by R i j = cos θ δ i j + ( 1 − cos θ ) n i n j − sin θ ϵ i j k n k . {\displaystyle R_{ij}=\cos \theta \,\delta _{ij}+(1-\cos \theta )n_{i}n_{j ...
Note how the use of A[i][j] with multi-step indexing as in C, as opposed to a neutral notation like A(i,j) as in Fortran, almost inevitably implies row-major order for syntactic reasons, so to speak, because it can be rewritten as (A[i])[j], and the A[i] row part can even be assigned to an intermediate variable that is then indexed in a separate expression.
In reverse Polish notation, [7] also known as postfix notation, all operations are entered after the operands on which the operation is performed. Reverse Polish notation is parenthesis-free, which usually leads to fewer button presses needed to perform an operation. By the use of a stack, one can enter formulas without the need to rearrange ...
Note that, since x + y is a vector and α is a multi-index, the expression on the left is short for (x 1 + y 1) α 1 ⋯(x n + y n) α n. Leibniz formula For smooth functions f {\textstyle f} and g {\textstyle g} , ∂ α ( f g ) = ∑ ν ≤ α ( α ν ) ∂ ν f ∂ α − ν g . {\displaystyle \partial ^{\alpha }(fg)=\sum _{\nu \leq \alpha ...
Thus the derivative of the Heaviside step function can be seen as the inward normal derivative at the boundary of the domain given by the positive half-line. In higher dimensions, the derivative naturally generalises to the inward normal derivative, while the Heaviside step function naturally generalises to the indicator function of some domain D.
Where needed, the notation extends to components of non-tensors, particularly multidimensional arrays. A tensor may be expressed as a linear sum of the tensor product of vector and covector basis elements. The resulting tensor components are labelled by indices of the basis. Each index has one possible value per dimension of the underlying ...