vectorialcalculus (version 1.0.5)

curl3d: Numerical curl of a three-dimensional vector field

Description

Computes the curl of a vector field in three dimensions at a given point, using second-order central finite differences. The field may optionally depend on a time parameter; if so, the curl is evaluated at a fixed time value.

Usage

curl3d(field, x, y, z, h = NULL, tval = 0, method = c("central"))

Value

A named numeric vector of length three containing the curl components at the evaluation point. The components are named omega_x, omega_y and omega_z.

Arguments

field

A function representing the vector field. It can be defined as function(x, y, z) or as function(x, y, z, t). It must return a numeric vector c(Fx, Fy, Fz).

x, y, z

Numeric scalars giving the coordinates of the evaluation point.

h

Step size for finite differences. It may be:

• a single numeric value,

• a numeric vector of length three,

• or NULL (automatic selection).

tval

Time value used when the vector field depends on time. Default is 0.

method

Differencing scheme. Currently only "central" is supported.

Details

The vector field must be a function that returns a numeric vector of length three representing the components of the field at a point. The curl is obtained by approximating the partial derivatives of the field components with respect to each coordinate direction using symmetric finite differences.

The step size for each coordinate can be:

• a single scalar used for all three axes,

• a numeric vector of length three providing separate steps for the x, y and z directions,

• or NULL, in which case automatic step sizes are chosen based on the evaluation point.

The method currently implemented is the second-order central differencing scheme. Smaller step sizes may provide more accurate results for rapidly varying fields, at the cost of increased sensitivity to floating-point error.

Examples

# Simple rotating field: curl is constant in the third component
field1 <- function(x, y, z) c(-y, x, 0.6)
curl3d(field1, x = 0.1, y = -0.3, z = 2)

# Time-dependent example (time does not affect the curl):
field2 <- function(x, y, z, t) c(-y, x + t, z)
curl3d(field2, x = 1, y = 2, z = 3, tval = 5)

# Using a smaller step size for more precision:
curl3d(field1, x = 1, y = 1, z = 1, h = 1e-5)