vignettes/integrals.Rmd
integrals.Rmd
The package integrates seamlessly with cubature for
efficient numerical integration in C
. The function integral
provides the interface for multidimensional integrals of
functions
, expressions
, and
characters
in arbitrary orthogonal
coordinate systems. If the package cubature is not
installed, the package implements a naive Monte Carlo integration by
default. The function returns a list
containing the
value
of the integral as well as other information on the
estimation uncertainty. The integration bounds are specified via the
argument bounds
: a list containing the lower and upper
bound for each variable. If the two bounds coincide, or if a single
number is specified, the corresponding variable is not integrated and
its value is fixed. For arbitrary orthogonal coordinates
the integral is computed as:
where is the Jacobian determinant of the transformation and is equal to the product of the scale factors .
Univariate integral :
that is equivalent to:
Univariate integral :
Multivariate integral :
Area of a circle
i <- integral(f = 1,
bounds = list(r = c(0,1), theta = c(0,2*pi)),
coordinates = "polar")
i$value
#> [1] 3.141593
Volume of a sphere
i <- integral(f = 1,
bounds = list(r = c(0,1), theta = c(0,pi), phi = c(0,2*pi)),
coordinates = "spherical")
i$value
#> [1] 4.188794
As a final example consider the electric potential in spherical coordinates arising from a unitary point charge:
V <- "1/(4*pi*r)"
The electric field is determined by the gradient of the potential1 :
Then, by Gauss’s law2, the total charge enclosed within a given volume is equal to the surface integral of the electric field where denotes the scalar product between the two vectors. In spherical coordinates, this reduces to the surface integral of the radial component of the electric field . The following code computes this surface integral on a sphere with fixed radius :
i <- integral(E[1],
bounds = list(r = 1, theta = c(0,pi), phi = c(0,2*pi)),
coordinates = "spherical")
i$value
#> [1] 1.000002
As expected , the unitary charge generating the electric potential.
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
A BibTeX entry for LaTeX users is