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

## Arguments

f

array of characters or a function returning a numeric array.

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 derivative.

params

list of additional parameters passed to f.

coordinates

coordinate system to use. One of: cartesian, polar, spherical, cylindrical, parabolic, parabolic-cylindrical or a vector of scale factors for each varibale.

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 TRUE, return the divergence as a scalar and not as an array for vector-valued functions.

## Value

Scalar for vector-valued functions when drop=TRUE, array otherwise.

## Details

The divergence of a vector-valued 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 vector-valued functions $$F_{d_1\dots d_n,i}$$, the divergence is computed for each vector:

$$(\nabla \cdot F)_{d_1\dots d_n} = \frac{1}{J}\sum_i\partial_i\Biggl(\frac{J}{h_i}F_{d_1\dots d_n,i}\Biggl)$$

## Functions

• %divergence%: binary operator with default parameters.

## References

Guidotti, E. (2020). "calculus: High dimensional numerical and symbolic calculus in R". https://arxiv.org/abs/2101.00086

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"))
#> [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

### vectorized interface
f <- function(x) c(x[1]^2, x[2]^3, x[3]^4)
divergence(f, var = c(1,1,1))
#> [1] 9

### symbolic array of vector-valued 3-d 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 vector-valued 3-d 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

### binary operator
c("x^2","y^3","z^4") %divergence% c("x","y","z")
#> [1] "2 * x + 3 * y^2 + 4 * z^3"