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Chapter 4: Partial Differentiation
Section 4.6: Surface Normal and Tangent Plane
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Example 4.6.1
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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 4.6.1(a) shows the surface in green, the tangent plane at Q: 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 , 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=[2,-3,-5/6],<4/3,-3,1>);
P:=65/6-4*x/3+3*y;
p1:=PlotVector(N,color=black,width=.4);
p2:=plot3d(P,x=0..4,y=-5..5,style=surface,color=red,transparency=.3);
p3:=plot3d(5-x^2/3-y^2/2,x=-4..4,y=-5..5,style=surface,color=green);
p4:=display(p1,p2,p3,scaling=constrained,axes=frame,view=-3..5,labels=[x,y,z],tickmarks=[4,8,4],orientation=[-120,75,0]);
print(p4);
end module:
end use:
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Figure 4.6.1(a) Surface, normal, and tangent plane
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and then by
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Maple Solution - Interactive
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A complete solution is available with the Student MultivariateCalculus package.
Let Q be the point on the surface that corresponds to the point .
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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 Function
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Context Panel: Assign Name
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Obtain a surface normal at point Q
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Matrix palette: Template for a 3-component vector
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Calculus palette: Partial differentiation operator
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Context Panel: Evaluate and Display Inline
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Context Panel: Evaluate at a Point≻
(See Figure 4.6.1(b).)
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Context Panel: Assign to a Name≻N
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Figure 4.6.1(b) Dialog: Evaluate at a Point
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Obtain an equation for the tangent plane
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Write a sequence of names for 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.
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Convert Q to the position vector A
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Write the name for point Q.
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Context Panel: Evaluate and Display Inline
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Context Panel: Conversions≻Column Vector
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Context Panel: Assign to a Name≻A
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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 diff and eval commands to obtain partial derivatives evaluated on the surface.
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Obtain an equation for the tangent plane
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Use the Plane command to generate the plane data-structure.
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The tangent plane can also be obtained via the TangentPlane command in the Student VectorCalculus package.
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The plane is given in the form of a position vector, where the third component is interpreted as the equation .
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<< Chapter Overview Section 4.6
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