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 vector-valued function 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 and the Levi-Civita symbol
epsilon
:
In 3 dimensions:
where . In
dimensions, the
curl
is implemented in such
a way that the formula reduces correctly to the previous cases for and
:
When is an
array
of vector-valued functions the
curl
is computed for each vector:
Value
Vector for vector-valued functions when drop=TRUE
, array
otherwise.
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
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"))
### 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))
### 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))
### 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 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"))
### 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))
### binary operator
c("x*y","y*z","x*z") %curl% c("x","y","z")