Chapter 4: Partial Differentiation
Section 4.6: Surface Normal and Tangent Plane
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Example 4.6.4
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At P: on the surface defined by , obtain and draw both the normal and tangent plane.
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Solution
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Mathematical Solution
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Figure 4.6.4(a) shows the surface in green, the tangent plane at P: in red, and the normal at this point in black.
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According to Table 4.6.1, N is obtained by evaluating at P, yielding
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The tangent plane is then given vectorially by
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use plots, Student:-VectorCalculus in
module()
local P,N,p1,p2,p3,p4;
N:=RootedVector(root=[1,1,1],<2,4,6>);
P:=(6-x-2*y)/3;
p1:=PlotVector(N,color=black,width=.4);
p2:=plot3d(P,x=-2..2,y=-2..4,style=surface,color=red,transparency=.3);
p3:=implicitplot3d(x^2+2*y^2+3*z^2-6,x=-3..3,y=-2..2,z=-2..2,style=surface,color=green);
p4:=display(p1,p2,p3,scaling=constrained,axes=frame,view=-2..7,labels=[x,y,z],tickmarks=[4,4,4],orientation=[-15,80,0]);
print(p4);
end module:
end use:
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Figure 4.6.4(a) Surface, normal, and tangent plane
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and then by
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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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Obtain a surface normal at point P
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Context Panel: Student Multivariate Calculus≻Differentiate≻Gradient
Evaluate at P (see Figure 4.6.4(c).
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Context Panel: Select Element≻1
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Context Panel: Assign to a Name≻N
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Figure 4.6.6(c) Dialog: Evaluate at a Point
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Obtain an equation for the tangent plane
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Write a sequence of the point and normal that define the tangent plane.
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Context Panel: Student Multivariate Calculus≻Lines & Planes≻Plane
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Context Panel: Student Multivariate Calculus≻Lines & Planes≻Representation
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Maple also supports a solution from first principles.
Represent point P as the position vector A
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Context Panel: Assign Name
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Define the generic position vector R and implement the vector equation of a plane
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Context Panel: Assign Name
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Write the vector equation of the plane that has normal N and passes through point A.
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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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Obtain a vector normal to the surface
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Use the Gradient command to obtain, at P, the gradient of .
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=
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Obtain an equation for the tangent plane
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The tangent plane can also be obtained via the TangentPlane command in the Student VectorCalculus package.
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