Gradient - Maple Help

Student[MultivariateCalculus]

 return the gradient at specified points

 Calling Sequence Gradient(f(x,y,..), [x,y,..]=[a1,b1,..], x=xmin..xmax, y=ymin..ymax, ..., opts) Gradient(f(x,y,..), [x,y,..]=[[a1,b1,..],[a2,b2,..],..], x=xmin..xmax, y=ymin..ymax, ..., opts)

Parameters

 f(x, y, ..) - algebraic expression x, y, .. - name; independent variables [a1, b1, ..] - list or list of lists of real constants or names; point(s) at which the gradient is evaluated xmin, xmax, ymin, ymax, ... - (optional) real constants specifying range over which to plot function opts - (optional) equation(s) of the form option=value where option is one of functionoptions, gradientoptions, curveoptions, output, showfunction, showcurves, showprojectedcurves, gradplotoptions, title, caption or view; specify output and/or plot options

Description

 • The Gradient(f(x, y,...), [x, y,...] = [a1, b1,...]) calling sequence returns the gradient of the function f at specified points. Multiple points can be given by putting them into a list, for example, Gradient(f(x,y), [x,y] = [[1,2],[2,0]]). In the case of a function of 2 variables, the output option can be used to request that a plot or gradplot be returned.
 • The Del and Nabla commands are both aliases for the Gradient command.
 • The GradientTutor routine offers equivalent capabilities to Gradient in a tutor interface.  See the Student[MultivariateCalculus][GradientTutor] help page.
 • If output=plot is given (so in particular, the function expression has at most two variables), the opts argument can be used to control the appearance of the plot.
 caption = string
 Specifies the plot or gradplot caption.  The default caption is constructed from the parameters.
 functionoptions = list
 Specifies the plot options for plotting the function $f\left(x,y\right)$. For more information on plotting options, see plot3d/options.
 curveoptions = list
 Specifies the plot options for plotting the level curve of the function at the point where the gradient is computed, and its projection on the x-y plane. For more information on plotting options, see plot3d/options.
 Specifies the plot options for plotting the gradient of the function. For more information on plotting options, see plot3d/options.
 output = value, plot, or gradplot
 This option controls the return value of the function.
 * output = value specifies that the value of the gradient is to be returned. Plot options are ignored if output = value.  The default is output = value.
 * output = plot specifies that a plot displays, which shows the function, a level curve at the point, the projection of that level curve onto the x-y plane, and a scaled vector in the direction of the gradient.
 * output = gradplot specifies that a gradplot displays.
 Specifies the plot options if the output is a gradplot. For more information on plotting options, see plot/options.
 showfunction = true or false
 Determines whether the function $f\left(x,y\right)$ is plotted. The default is true.
 showcurves = true or false
 Determines whether the level curve is plotted. The default is true.
 showprojectedcurves = true or false
 Determines whether the projection of the level curve is plotted; this option is ignored if showcurves is false. The default is true.
 For more information on typesetting the title or caption, see plot/typesetting.
 title = anything
 A title for the plot.
 The default title is constructed from the parameters and the command options. title = "" disables the default title. For more information about specifying a title, see plot/typesetting.
 caption = anything
 A caption for the plot.
 The default caption is constructed from the parameters and the command options. caption = "" disables the default caption. For more information about specifying a caption, see plot/typesetting.
 view = list
 Specifies the plot or gradplot view.
 • For information on how to change the default colors, see the Student[SetColors] help page.

Examples

 > $\mathrm{with}\left(\mathrm{Student}\left[\mathrm{MultivariateCalculus}\right]\right):$
 > $\mathrm{Gradient}\left({x}^{2}+{y}^{2},\left[x,y\right]=\left[0,1\right]\right)$
 $\left[\left[\begin{array}{c}{0}\\ {2}\end{array}\right]\right]$ (1)

The Del and Nabla commands are both aliases for Gradient:

 > $\mathrm{Del}\left({x}^{2}+{y}^{2},\left[x,y\right]=\left[0,1\right]\right)$
 $\left[\left[\begin{array}{c}{0}\\ {2}\end{array}\right]\right]$ (2)
 > $\nabla \left({x}^{2}+{y}^{2},\left[x,y\right]=\left[0,1\right]\right)$
 $\left[\left[\begin{array}{c}{0}\\ {2}\end{array}\right]\right]$ (3)
 > $\mathrm{Gradient}\left({x}^{2}+{y}^{2},\left[x,y\right]=\left[\left[0,1\right],\left[2,1\right]\right]\right)$
 $\left[\left[\begin{array}{c}{0}\\ {2}\end{array}\right]{,}\left[\begin{array}{c}{4}\\ {2}\end{array}\right]\right]$ (4)
 > $\mathrm{Gradient}\left({x}^{2}+{y}^{2},\left[x,y\right]=\left[\left[0,1\right],\left[2,1\right]\right],\mathrm{output}=\mathrm{gradplot}\right)$

The command to create the plot from the Plotting Guide is

 > $\mathrm{Gradient}\left({x}^{2}+{y}^{2},\left[x,y\right]=\left[\left[0,1\right],\left[2,1\right]\right],x=-1..6,y=-1..6,z=0..10,\mathrm{output}=\mathrm{plot}\right)$

Compatibility

 • The curveoptions, showcurves and showprojectedcurves options were introduced in Maple 17.
 • For more information on Maple 17 changes, see Updates in Maple 17.