Computes the numerical divergence of functions
or the symbolic divergence of characters
in arbitrary orthogonal coordinate systems.
divergence( f, var, params = list(), coordinates = "cartesian", accuracy = 4, stepsize = NULL, drop = TRUE ) f %divergence% var
f  array of 

var  vector giving the variable names with respect to which the derivatives are to be computed and/or the point where the derivatives are to be evaluated. See 
params 

coordinates  coordinate system to use. One of: 
accuracy  degree of accuracy for numerical derivatives. 
stepsize  finite differences stepsize for numerical derivatives. It is based on the precision of the machine by default. 
drop  if 
Scalar for vectorvalued functions when drop=TRUE
, array
otherwise.
The divergence of a vectorvalued function \(F_i\) produces a scalar value
\(\nabla \cdot F\) representing the volume density of the outward flux of the
vector field from an infinitesimal volume around a given point.
The divergence
is computed in arbitrary orthogonal coordinate systems using the
scale factors \(h_i\):
$$\nabla \cdot F = \frac{1}{J}\sum_i\partial_i\Biggl(\frac{J}{h_i}F_i\Biggl)$$
where \(J=\prod_ih_i\). When \(F\) is an array
of vectorvalued functions
\(F_{d_1\dots d_n,i}\), the divergence
is computed for each vector:
$$(\nabla \cdot F)_{d_1\dots d_n} = \frac{1}{J}\sum_i\partial_i\Biggl(\frac{J}{h_i}F_{d_1\dots d_n,i}\Biggl)$$
%divergence%
: binary operator with default parameters.
Guidotti, E. (2020). "calculus: High dimensional numerical and symbolic calculus in R". https://arxiv.org/abs/2101.00086
### symbolic divergence of a vector field f < c("x^2","y^3","z^4") divergence(f, var = c("x","y","z"))#> [1] "2 * x + 3 * y^2 + 4 * z^3"### numerical divergence of a vector field in (x=1, y=1, z=1) f < function(x,y,z) c(x^2, y^3, z^4) divergence(f, var = c(x=1, y=1, z=1))#> [1] 9#> [1] 9### symbolic array of vectorvalued 3d functions f < array(c("x^2","x","y^2","y","z^2","z"), dim = c(2,3)) divergence(f, var = c("x","y","z"))#> [1] "2 * x + 2 * y + 2 * z" "1 + 1 + 1"### numeric array of vectorvalued 3d functions in (x=0, y=0, z=0) f < function(x,y,z) array(c(x^2,x,y^2,y,z^2,z), dim = c(2,3)) divergence(f, var = c(x=0, y=0, z=0))#> [1] 0 3#> [1] "2 * x + 3 * y^2 + 4 * z^3"