Chapter 4: Partial Differentiation
Section 4.5: Gradient Vector
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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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Solution
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Mathematical Solution
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Part (a):
= =
Part (b): See Figure 4.5.2.(a).
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use plots, Student:-VectorCalculus in
module()
local V,p1,p2,p3;
V:=RootedVector(root=[1,1,1],<2,4,6>);
p1:=PlotVector(V,color=black);
p2:=implicitplot3d(x^2+2*y^2+3*z^2=6,x=-3..3,y=-2..2,z=-2..2,style=surface);
p3:=display(p1,p2,scaling=constrained,axes=frame,tickmarks=[6,8,8],orientation=[45,80,0]);
print(p3);
end module:
end use:
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Figure 4.5.2(a) Level surface and gradient
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Part (c):
The coordinate curves
and
lie in the surface defined by , pass through the point P, and project respectively to the grid lines and , in the -plane.
At P on the surface, the vectors
and
are tangent to the curves and , respectively. That the gradient vector is orthogonal to the coordinates curves and is shown by the vanishing of the dot products, that is, by the calculations
= and =
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Maple Solution - Interactive
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Initialize
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Tools≻Load Package: Student Multivariate Calculus
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Loading Student:-MultivariateCalculus
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Context Panel: Assign name
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Part (a)
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Write .
Context Panel: Evaluate and Display Inline
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Context Panel: Student Multivariate Calculus≻Differentiate≻Gradient
(See Figure 4.5.2(b).)
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Context Panel: Select Element≻1
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Context Panel: Assign to a Name≻Gw
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Figure 4.5.2(b) Gradient dialog
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Part (b)
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Type and press the Enter key.
Context Panel: Plots≻Arrow from point
See dialog in Figure 4.5.2(c).
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Figure 4.5.2(c) Dialog for graphing an arrow
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Type and press the Enter key.
Context Panel: Plots≻Plot Builder
Obtain a graph of the surface .
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Control-drag (or copy/paste) the arrow onto the graph of the surface.
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Part (c)
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Write the equation .
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Context Panel: Evaluate and Display Inline
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Context Panel: Solve≻Obtain Solutions for≻
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Context Panel: Select Element≻1
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Context Panel: Assign to a Name≻Z
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= ⇒
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Expression palette: Evaluation template
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Context Panel: Evaluate and Display Inline
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Context Panel: Assign to a Name≻R[1] or R[2], as appropriate.
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=
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=
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Expression palette: Evaluation template
Calculus palette: Differentiation operator
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Context Panel: Evaluate and Display Inline
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Context Panel: Simplify≻Simplify
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Context Panel: Assign to a Name≻T[1] or T[2], as appropriate.
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=
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=
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Common Symbols palette: Dot product operator
Context Panel: Evaluate and Display Inline
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=
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Common Symbols palette: Dot product operator
Context Panel: Evaluate and Display Inline
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=
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Maple Solution - Coded
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Initialize
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Install the Student MultivariateCalculus package.
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Define .
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Part (a)
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Use the Gradient command to obtain and the eval command to obtain .
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Part (b)
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Use the RootedVector command to define with its "tail" at P:.
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Use the implicitplot3d command to graph the level surface .
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Use the display command to merge the graphs of the surface and its gradient.
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Part (c)
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Obtain by applying the solve command to .
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Use the eval command to obtain in the position-vector form of the plane section .
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Use the eval command to obtain in the position-vector form of the plane section .
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Use the simplify, eval, and diff commands to obtain the tangent vector .
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Use the simplify, eval, and diff commands to obtain the tangent vector .
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=
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=
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