Specifically for these equations, for any choice of a twice-differentiable scalar function of position and time " ? ", if is a solution for a given system, then so is another potential given by:
42.
In terms of the original scalar function, what this means is that every smooth surface can be approximated locally to second order by an ellipsoid ( just as it can be approximated locally to first order by a plane ).
43.
If the vector field \ mathbf { u } is known, the above equation can be solved for the scalar function \, \ phi and the divergence-free part of \ mathbf { u } can be extracted using the relation
44.
Where " " is the divergence operator ( also symbolized " div " ) which maps vector functions to scalar functions, and " is the gradient operator ( also symbolized " grad " ) which maps scalar functions to vector functions.
45.
Where " " is the divergence operator ( also symbolized " div " ) which maps vector functions to scalar functions, and " is the gradient operator ( also symbolized " grad " ) which maps scalar functions to vector functions.
46.
This form is called " exact " on a domain D \ subset \ mathbb { R } ^ 3 in space if there exists some scalar function Q = Q ( x, y, z ) defined on D such that
47.
In the multivariate case where \ mathbf { x } is a d-dimensional vector and f ( \ mathbf { x } ) is a scalar function of \ mathbf { x }, Laplace's approximation is usually written as:
48.
Generalization to vector-valued functions ( a, b ) \ to \ mathbb { R } ^ d is straightforward; one applies the results for scalar functions to each coordinate separately, or treats the vector-valued case from the beginning.
49.
Then the * reflection line function * \ theta ( p ) : M \ rightarrow (-\ pi, \ pi ] is a scalar function mapping points on the surface to angles between v _ a and the projected reflected view direction d:
50.
In addition, gradient of a scalar function ( as soon as the space is assumed to be simply connected, see "'Note 1 "'below ), and that a divergenceless field can be written as a curl of another field.
