Computes the numerical Jacobian of functions or the symbolic Jacobian of characters in arbitrary orthogonal coordinate systems.

jacobian(
f,
var,
params = list(),
coordinates = "cartesian",
accuracy = 4,
stepsize = NULL
)

f %jacobian% var

## Arguments

f

array of characters or a function returning a numeric array.

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 derivative.

params

list of additional parameters passed to f.

coordinates

coordinate system to use. One of: cartesian, polar, spherical, cylindrical, parabolic, parabolic-cylindrical or a vector of scale factors for each varibale.

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.

## Value

array.

## Details

The function is basically a wrapper for gradient with drop=FALSE.

## Functions

• f %jacobian% 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

Other differential operators: curl(), derivative(), divergence(), gradient(), hessian(), laplacian()

## Examples

### symbolic Jacobian
jacobian("x*y*z", var = c("x", "y", "z"))
#>      [,1]    [,2]    [,3]
#> [1,] "y * z" "x * z" "x * y"

### numerical Jacobian in (x=1, y=2, z=3)
f <- function(x, y, z) x*y*z
jacobian(f = f, var = c(x=1, y=2, z=3))
#>      [,1] [,2] [,3]
#> [1,]    6    3    2

### vectorized interface
f <- function(x) x[1]*x[2]*x[3]
jacobian(f = f, var = c(1, 2, 3))
#>      [,1] [,2] [,3]
#> [1,]    6    3    2

### symbolic vector-valued functions
f <- c("y*sin(x)", "x*cos(y)")
jacobian(f = f, var = c("x","y"))
#>      [,1]         [,2]
#> [1,] "y * cos(x)" "sin(x)"
#> [2,] "cos(y)"     "-(x * sin(y))"

### numerical vector-valued functions
f <- function(x) c(sum(x), prod(x))
jacobian(f = f, var = c(0,0,0))
#>      [,1] [,2] [,3]
#> [1,]    1    1    1
#> [2,]    0    0    0

### binary operator
"x*y^2" %jacobian% c(x=1, y=3)
#>      [,1] [,2]
#> [1,]    9    6