curl {calculus}  R Documentation 
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 
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 
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)
Vector for vectorvalued functions when drop=TRUE
, array
otherwise.
f %curl% var
: binary operator with default parameters.
Guidotti E (2022). "calculus: HighDimensional Numerical and Symbolic Calculus in R." Journal of Statistical Software, 104(5), 137. doi:10.18637/jss.v104.i05
Other differential operators:
derivative()
,
divergence()
,
gradient()
,
hessian()
,
jacobian()
,
laplacian()
### 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")