Computes the numerical curl of functions or the symbolic curl of characters in arbitrary orthogonal coordinate systems.

curl(
  f,
  var,
  params = list(),
  coordinates = "cartesian",
  accuracy = 4,
  stepsize = NULL,
  drop = TRUE
)

f %curl% 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 curl as a vector and not as an array for vector-valued functions.

Value

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

Details

The curl of a vector-valued function \(F_i\) at a point is represented by a vector whose length and direction denote the magnitude and axis of the maximum circulation. In 2 dimensions, the curl is computed in arbitrary orthogonal coordinate systems using the scale factors \(h_i\) and the Levi-Civita symbol epsilon:

$$\nabla \times F = \frac{1}{h_1h_2}\sum_{ij}\epsilon_{ij}\partial_i\Bigl(h_jF_j\Bigl)= \frac{1}{h_1h_2}\Biggl(\partial_1\Bigl(h_2F_2\Bigl)-\partial_2\Bigl(h_1F_1\Bigl)\Biggl)$$

In 3 dimensions:

$$(\nabla \times F)_k = \frac{h_k}{J}\sum_{ij}\epsilon_{ijk}\partial_i\Bigl(h_jF_j\Bigl)$$

where \(J=\prod_i h_i\). In \(m+2\) dimensions, the curl is implemented in such a way that the formula reduces correctly to the previous cases for \(m=0\) and \(m=1\):

$$(\nabla \times F)_{k_1\dots k_m} = \frac{h_{k_1}\cdots h_{k_m}}{J}\sum_{ij}\epsilon_{ijk_1\dots k_m}\partial_i\Bigl(h_jF_j\Bigl)$$

When \(F\) is an array of vector-valued functions \(F_{d_1,\dots,d_n,j}\) the curl is computed for each vector:

$$(\nabla \times F)_{d_1\dots d_n,k_1\dots k_m} = \frac{h_{k_1}\cdots h_{k_m}}{J}\sum_{ij}\epsilon_{ijk_1\dots k_m}\partial_i\Bigl(h_jF_{d_1\dots d_n,j}\Bigl)$$

Functions

  • %curl%: binary operator with default parameters.

References

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

See also

Other differential operators: derivative(), divergence(), gradient(), hessian(), jacobian(), laplacian()

Examples

### symbolic curl of a 2-d vector field f <- c("x^3*y^2","x") curl(f, var = c("x","y"))
#> [1] "(1) * 1 + (x^3 * (2 * y)) * -1"
### numerical curl of a 2-d vector field in (x=1, y=1) f <- function(x,y) c(x^3*y^2, x) curl(f, var = c(x=1, y=1))
#> [1] -1
### numerical curl of a 3-d vector field in (x=1, y=1, z=1) f <- function(x,y,z) c(x^3*y^2, x, z) curl(f, var = c(x=1, y=1, z=1))
#> [1] 0 0 -1
### vectorized interface f <- function(x) c(x[1]^3*x[2]^2, x[1], x[3]) curl(f, var = c(1,1,1))
#> [1] 0 0 -1
### symbolic array of vector-valued 3-d functions f <- array(c("x*y","x","y*z","y","x*z","z"), dim = c(2,3)) curl(f, var = c("x","y","z"))
#> [,1] [,2] [,3] #> [1,] "(y) * -1" "(z) * -1" "(x) * -1" #> [2,] "0" "0" "0"
### numeric array of vector-valued 3-d functions in (x=1, y=1, z=1) f <- function(x,y,z) array(c(x*y,x,y*z,y,x*z,z), dim = c(2,3)) curl(f, var = c(x=1, y=1, z=1))
#> [,1] [,2] [,3] #> [1,] -1 -1 -1 #> [2,] 0 0 0
### binary operator c("x*y","y*z","x*z") %curl% c("x","y","z")
#> [1] "(y) * -1" "(z) * -1" "(x) * -1"