Chapter 4: Partial Differentiation
Section 4.5: Gradient Vector
The gradient of fx,y is the vector ∇f=fx i+fy j, whereas the gradient of fx,y,z is the vector ∇f=fx i+fy j+fz k. Table 5.4.1 lists the five most important properties of the gradient vector.
The gradients of fx,y are orthogonal to the level curves y=yx defined implicitly by fx,y=c, where c is a real constant.
The gradients of fx,y,z are orthogonal to the level surfaces z=zx,y defined implicitly by fx,y,z=c, where c is a real constant.
At any point where ∇f≠0, the gradient ∇f points in the direction of increasing values of f.
Where ∇f≠0, it necessarily points in the direction of maximal increase in f.
The maximal rate of change in f, as measured by the directional derivative, has the value ∇f.
Table 5.4.1 Properties of the gradient vector
Obtaining the Gradient in Maple
The Del (or Nabla) operator ∇ =i ∂∂x+j ∂∂y+k ∂∂z, applied to a scalar fx,y,z, results in the gradient vector ∇f=fx i+fy j+fz k. Maple has ∇, the Nabla symbol, in its Common Symbols palette, but it only works as an operator when one of the VectorCalculus packages is loaded.
With the Student VectorCalculus package loaded, the ∇-operator will correctly compute the gradient in a Cartesian frame for any expression in either two or three names of coordinate variables. Thus, ∇x+y would correctly return i+j, but applied to a+x, would treat both a and x as independent variables, and differentiate with respect to both. The Gradient command itself admits a list of names with respect to which differentiation is to take place, but this list cannot be made know to the ∇-operator without the use of the SetCoordinates command. Finally, note that the gradient is returned as a VectorField, a more complex data structure than a "free vector" such as x,y.
The Gradient command in the Student MultivariateCalculus package never links to the Nabla symbol, and always requires as an additional parameter, a list of the variables of differentiation. However, this list of names can be set equal to a list of coordinates, so that the gradient is computed at a specific point.
Of course, the gradient of, say, fx,y, could also be obtained by the construct ∂∂ x f,∂∂ y f.
With the Student MultivariateCalculus package loaded, the Context Panel, launched on an expression in two or three variables, provides interactive access to the Gradient command.
There is also a Gradient command in the Physics:-Vectors package, but it only acts on vectors defined within that package. No use is made of that package, or its structures, in this work.
Let fx,y=5−2 x2−3 y2 and let P be the point 1,2.
Obtain ∇f at P.
Graph the surface z=fx,y.
On the same set of axes, graph the level curve through P, and ∇f at P.
At P, show that ∇f is orthogonal to a vector tangent to the level curve through P.
At P, obtain ψ=∇f·u, the directional derivative of f in the direction u=cost i+sint j . Show that ψ is a maximum when u is along ∇fP and that this maximum is ∇fP2.
Let w=x2+2 y2+3 z2 and let P be the point 1,1,1.
Obtain ∇w at P.
On the same set of axes, graph the level surface w=6 and ∇w at P.
At P, show that ∇w is orthogonal to the level surface w=6. Hint: Show that this gradient is orthogonal to the x- and y-coordinate curves through P.
Prove Property 1 in Table 4.5.1.
Prove Property 2 in Table 4.5.1.
Prove Property 3 in Table 4.5.1.
Prove Property 4 in Table 4.5.1.
Prove Property 5 in Table 4.5.1.
Show both graphically and analytically that the level curves of u=x2−y2−3 x+2 are orthogonal to the level curves of v=2⁢x⁢y−3 y.
Show both graphically and analytically that the level curves of u=sin⁡x⁢cosh⁡y are orthogonal to the level curves of v=cos⁡x⁢sinh⁡y.
If u=x4+2⁢x2⁢y2+y4−1x2+y2+2⁢y+1⁢x2+y2−2⁢y+1 and v=4⁢x⁢yx2+y2+2⁢y+1⁢x2+y2−2⁢y+1, show both graphically and analytically that their level curves are mutually orthogonal.
At P:2,1, determine the maximal rate of change and its direction for fx,y=x yx2+y3.
At P:1,2,3, determine the maximal rate of change and its direction for fx,y,z=x2−2 y+5 zx+y z.
<< Previous Section Table of Contents Next Section >>
© Maplesoft, a division of Waterloo Maple Inc., 2023. All rights reserved. This product is protected by copyright and distributed under licenses restricting its use, copying, distribution, and decompilation.
For more information on Maplesoft products and services, visit www.maplesoft.com
Download Help Document