Numerical and Symbolic Divergence

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

f %divergence% var: binary operator with default parameters.

References

Guidotti E (2022). "calculus: High-Dimensional Numerical and Symbolic Calculus in R." Journal of Statistical Software, 104(5), 1-37. doi:10.18637/jss.v104.i05

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"