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
```

- 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.

Vector for vector-valued functions when `drop=TRUE`

, `array`

otherwise.

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:

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

`%curl%`

: binary operator with default parameters.

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

Other differential operators:
`derivative()`

,
`divergence()`

,
`gradient()`

,
`hessian()`

,
`jacobian()`

,
`laplacian()`

```
### 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"
```