Chapter 4: Partial Differentiation
Section 4.5: Gradient Vector
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Essentials
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The gradient of is the vector , whereas the gradient of is the vector . Table 5.4.1 lists the five most important properties of the gradient vector.
Property
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Description
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1
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The gradients of are orthogonal to the level curves defined implicitly by , where is a real constant.
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2
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The gradients of are orthogonal to the level surfaces defined implicitly by , where is a real constant.
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3
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At any point where , the gradient points in the direction of increasing values of .
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4
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Where , it necessarily points in the direction of maximal increase in .
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5
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The maximal rate of change in , as measured by the directional derivative, has the value .
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Table 5.4.1 Properties of the gradient vector
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Obtaining the Gradient in Maple
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The Del (or Nabla) operator , applied to a scalar , results in the gradient vector . 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.
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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, would correctly return , but applied to , would treat both and 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 .
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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.
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Of course, the gradient of, say, , could also be obtained by the construct .
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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.
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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.
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Examples
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Example 4.5.1
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Let and let P be the point .
b)
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Graph the surface .
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c)
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On the same set of axes, graph the level curve through P, and at P.
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d)
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At P, show that is orthogonal to a vector tangent to the level curve through P.
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e)
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At P, obtain , the directional derivative of in the direction . Show that is a maximum when u is along and that this maximum is .
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Example 4.5.2
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Let and let P be the point .
b)
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On the same set of axes, graph the level surface and at P.
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c)
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At P, show that is orthogonal to the level surface . Hint: Show that this gradient is orthogonal to the - and -coordinate curves through P.
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Example 4.5.3
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Prove Property 1 in Table 4.5.1.
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Example 4.5.4
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Prove Property 2 in Table 4.5.1.
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Example 4.5.5
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Prove Property 3 in Table 4.5.1.
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Example 4.5.6
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Prove Property 4 in Table 4.5.1.
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Example 4.5.7
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Prove Property 5 in Table 4.5.1.
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Example 4.5.8
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Show both graphically and analytically that the level curves of are orthogonal to the level curves of .
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Example 4.5.9
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Show both graphically and analytically that the level curves of are orthogonal to the level curves of .
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Example 4.5.10
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If and , show both graphically and analytically that their level curves are mutually orthogonal.
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Example 4.5.11
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At P:, determine the maximal rate of change and its direction for .
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Example 4.5.12
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At P:, determine the maximal rate of change and its direction for .
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