While expansions of scalar functions are by far the most common application of multipole expansions, they may also be generalized to describe tensors of arbitrary rank.
22.
Where \ mathbf { n } denotes the ( typically exterior ) boundary \ partial \ Omega and " f " is a given scalar function.
23.
In axisymmetric flow another stream function formulation, called the Stokes stream function, can be used to describe the velocity components of an incompressible flow with one scalar function.
24.
If connection of interest is Levi-Civita connection one can find a convenient formula for Laplacian of scalar function in terms of partial derivatives with respect to chosen coordinates:
25.
Where \ phi is scalar function, | g | is absolute value of determinant of metric ( the use of absolute value is necessary in inverse of the metric tensor
26.
For a single element system of atoms, three scalar functions must be specified : the embedding function, a pair-wise interaction, and an electron cloud contribution function.
27.
For the sake of the simplicity, in dealing with Heaviside and Dirac delta functions in a two-dimensional coordinate space, consider a scalar function d, defined as:
28.
For scalar functions, the derivative in the Volterra system is the logarithmic derivative, and so the Volterra system is not a multiplicative calculus and is not a non-Newtonian calculus.
29.
Formally, a Hamiltonian system is a dynamical system completely described by the scalar function H ( \ boldsymbol { q }, \ boldsymbol { p }, t ), the Hamiltonian.
30.
Up to an overall sign, the Laplace de Rham operator is equivalent to the previous definition of the Laplace Beltrami operator when acting on a scalar function; see the proof for details.
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