Numerical and Symbolic Curl

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

f %curl% 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 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"