Chapter 4: Partial Differentiation
Section 4.6: Surface Normal and Tangent Plane
At P:1,1,1 on the surface defined by fx,y,z≡x2+2 y2+3 z2=6, obtain and draw both the normal and tangent plane.
Figure 4.6.4(a) shows the surface in green, the tangent plane at P:1,1,1 in red, and the normal at this point in black.
According to Table 4.6.1, N is obtained by evaluating ∇f at P, yielding
The tangent plane is then given vectorially by
use plots, Student:-VectorCalculus in
Figure 4.6.4(a) Surface, normal, and tangent plane
and then by
=2 x+4 y+6 z−2−4−6
=2 x+4 y+6 z−12
Maple Solution - Interactive
Tools≻Load Package: Student Multivariate Calculus
Obtain a surface normal at point P
Context Panel: Student Multivariate Calculus≻Differentiate≻Gradient
Evaluate at P (see Figure 4.6.4(c).
Context Panel: Select Element≻1
Context Panel: Assign to a Name≻N
Figure 4.6.6(c) Dialog: Evaluate at a Point
x2+2 y2+3 z2→gradient246→select entry 1246→assign to a nameN
Obtain an equation for the tangent plane
Write a sequence of the point and normal that define the tangent plane.
Context Panel: Student Multivariate Calculus≻Lines & Planes≻Plane
Context Panel: Student Multivariate Calculus≻Lines & Planes≻Representation
1,1,1,N→make plane<< Plane 1 >>→representationx+2⁢y+3⁢z=6
Maple also supports a solution from first principles.
Represent point P as the position vector A
Context Panel: Assign Name
Define the generic position vector R and implement the vector equation of a plane
Write the vector equation of the plane that has normal N and passes through point A.
Press the Enter key.
Maple Solution - Coded
Install the Student MultivariateCalculus package.
Define the function f.
f≔x2+2 y2+3 z2:
Obtain a vector normal to the surface
Use the Gradient command to obtain, at P, the gradient of f.
N≔Gradientf,x,y,z=1,1,1 = 246
Use the Plane and GetRepresentation commands.
The tangent plane can also be obtained via the TangentPlane command in the Student VectorCalculus package.
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