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Chapter 4: Partial Differentiation
Section 4.5: Gradient Vector
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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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Solution
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Mathematical Solution
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Figure 4.5.8(a) shows the level curves of in black, and the level curves of in red. From the figure, it appears that these sets of level curves intersect at right angles.
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Analytically, the orthogonality of these sets of level curves is shown by the computation
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which shows that the respective gradient vectors are orthogonal. Since these gradient vectors are themselves orthogonal to vectors tangent to the level curves, the sets of level curves are then orthogonal to each other.
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>
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use plots in
module()
local u,v,p1,p2,p3;
u:=x^2-y^2-3*x+2;
v:=2*x*y-3*y;
p1:=contourplot(u,x=-3..6,y=-3..3,color=black,colorbar=false);
p2:=contourplot(v,x=-3..6,y=-3..3,color=red,colorbar=false);
p3:=display(p1,p2,scaling=constrained);
print(p3);
end module:
end use:
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Figure 4.5.8(a) Level curves of 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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Control-drag expressions for each of u and v.
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Context Panel: Student Multivariate Calculus≻Differentiate≻Gradient
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Context Panel: Assign to a Name≻Gu or Gv, as appropriate
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Show
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Common Symbols palette: Dot product operator
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Context Panel: Evaluate and Display Inline
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=
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Use the Plot Builder to generate Figure 4.5.8(a)
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Context Panel: Plot Builder≻2-D contour plot
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Color Options≻color 1≻black
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Context Panel: Plot Builder≻2-D contour plot
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Color Options≻color 1≻red
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Copy the graph on the left and paste it onto the one on the right.
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Maple Solution - Coded
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Initialize
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Install the Student VectorCalculus package.
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Show
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=
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Obtain Figure 4.5.8(a)
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The contourplot and display commands from the plots package will produce Figure 4.5.8(a), as illustrated by the following code. Simply remove the colon from the last line and execute.
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