vectorialcalculus (version 1.0.5)

partial_derivatives_surface: Partial derivatives of z = f(x, y) at a point with 3D visualization

Description

Numerically approximates the partial derivatives \(f_x(x_0, y_0)\) and \(f_y(x_0, y_0)\) of a scalar field \(z = f(x, y)\) at a given point \((x_0, y_0)\) using central finite differences.

Usage

partial_derivatives_surface(
  f,
  x0,
  y0,
  h = NULL,
  xlim = NULL,
  ylim = NULL,
  nx = 60L,
  ny = 60L,
  plot = TRUE,
  scene = list(aspectmode = "data", xaxis = list(title = "x"), yaxis = list(title = "y"),
    zaxis = list(title = "z")),
  bg = list(paper = "white", plot = "white")
)

Value

A list with components:

• fx: numeric scalar, approximation of \(f_x(x_0, y_0)\).

• fy: numeric scalar, approximation of \(f_y(x_0, y_0)\).

• f0: numeric scalar, \(f(x_0, y_0)\).

• fig: a plotly object if plot = TRUE and plotly is available; otherwise NULL.

Arguments

f

Function function(x, y) returning a numeric scalar f(x, y).

x0, y0

Numeric scalars; coordinates of the point where the partial derivatives are evaluated.

h

Numeric step for the central finite differences. If NULL, a default value is chosen as 1e-4 * (1 + max(abs(x0), abs(y0))).

xlim

Numeric length-2 vector c(x_min, x_max). Range for the \(x\)-axis used to draw the surface. If NULL, a symmetric window around x0 is used.

ylim

Numeric length-2 vector c(y_min, y_max). Range for the \(y\)-axis used to draw the surface. If NULL, a symmetric window around y0 is used.

nx, ny

Integer grid sizes (number of points) along x and y for the surface plot. Recommended values are at least 20.

plot

Logical; if TRUE, builds and returns a plotly surface plot. If FALSE, only the numeric derivatives are returned.

scene

List with plotly 3D scene options (axis titles, aspect mode, and so on) passed to plotly::layout().

bg

List with background colors for plotly, with components paper and plot.

Details

Optionally, it builds a 3D plotly surface for \(z = f(x, y)\) on a rectangular window around \((x_0, y_0)\) and overlays:

• the intersection curve of the surface with the plane \(y = y_0\) and its tangent line given by \(f_x(x_0, y_0)\);

• the intersection curve with the plane \(x = x_0\) and its tangent line given by \(f_y(x_0, y_0)\);

• the base point \((x_0, y_0, f(x_0, y_0))\).

Examples

if (interactive()) {
f <- function(x, y) x^2 + 3 * x * y - y^2
res <- partial_derivatives_surface(
  f,
  x0 = 1, y0 = -1,
  xlim = c(-1, 3),
  ylim = c(-3, 1),
  nx = 60, ny = 60
)
res$fx
res$fy
}