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→gradient
→select entry 1
→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.
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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