divergence {calculus}  R Documentation 
Numerical and Symbolic Divergence
Description
Computes the numerical divergence of functions
or the symbolic divergence of characters
in arbitrary orthogonal coordinate systems.
Usage
divergence(
f,
var,
params = list(),
coordinates = "cartesian",
accuracy = 4,
stepsize = NULL,
drop = TRUE
)
f %divergence% var
Arguments
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 
Details
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)
Value
Scalar for vectorvalued functions when drop=TRUE
, array
otherwise.
Functions

f %divergence% var
: binary operator with default parameters.
References
Guidotti E (2022). "calculus: HighDimensional Numerical and Symbolic Calculus in R." Journal of Statistical Software, 104(5), 137. doi:10.18637/jss.v104.i05
See Also
Other differential operators:
curl()
,
derivative()
,
gradient()
,
hessian()
,
jacobian()
,
laplacian()
Examples
### symbolic divergence of a vector field
f < c("x^2","y^3","z^4")
divergence(f, var = c("x","y","z"))
### 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))
### vectorized interface
f < function(x) c(x[1]^2, x[2]^3, x[3]^4)
divergence(f, var = c(1,1,1))
### 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"))
### 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))
### binary operator
c("x^2","y^3","z^4") %divergence% c("x","y","z")