curl {calculus}  R Documentation 
Numerical and Symbolic Curl
Description
Computes the numerical curl of functions
or the symbolic curl of characters
in arbitrary orthogonal coordinate systems.
Usage
curl(
f,
var,
params = list(),
coordinates = "cartesian",
accuracy = 4,
stepsize = NULL,
drop = TRUE
)
f %curl% var
Arguments
f 
array of 
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 
params 

coordinates 
coordinate system to use. One of: 
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 
Details
The curl of a vectorvalued 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 LeviCivita 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 vectorvalued 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)
Value
Vector for vectorvalued functions when drop=TRUE
, array
otherwise.
Functions

f %curl% var
: binary operator with default parameters.
References
Guidotti E (2022). "calculus: HighDimensional Numerical and Symbolic Calculus in R." Journal of Statistical Software, 104(5), 137. doi:10.18637/jss.v104.i05
See Also
Other differential operators:
derivative()
,
divergence()
,
gradient()
,
hessian()
,
jacobian()
,
laplacian()
Examples
### symbolic curl of a 2d vector field
f < c("x^3*y^2","x")
curl(f, var = c("x","y"))
### numerical curl of a 2d 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))
### numerical curl of a 3d 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))
### vectorized interface
f < function(x) c(x[1]^3*x[2]^2, x[1], x[3])
curl(f, var = c(1,1,1))
### symbolic array of vectorvalued 3d functions
f < array(c("x*y","x","y*z","y","x*z","z"), dim = c(2,3))
curl(f, var = c("x","y","z"))
### numeric array of vectorvalued 3d 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))
### binary operator
c("x*y","y*z","x*z") %curl% c("x","y","z")