Chapter 4: Partial Differentiation
Section 4.6: Surface Normal and Tangent Plane
At P:a,b on the surface defined by z=fx,y, obtain an equation for the tangent plane in the form z=….
If A is the position-vector representation of the point of contact of the plane on the surface; R, the generic position vector to the arbitrary point x,y,z; and N, a surface normal at the point of contact, then the vector equation for the tangent plane is R−A·N=0. This leads to
which can be rearranged to
Maple Solution - Interactive
Guided by the Mathematical Solution, enter the appropriate equation, press the Enter key, and use the Context Panel's Solve option to isolate z.
Tools≻Load Package: Student Multivariate Calculus
→isolate for z
Tangent plane as first-degree Taylor polynomial
Alternatively, recognize the expression for tangent plane as the first-degree Taylor polynomial. Obtain this interactively via
Context Panel: Series≻Multivariate Taylor Polynomial.
(Complete the resulting pop-up dialog as per Figure 4.6.2(a).)
Figure 4.6.2(a) Taylor Polynomial dialog
Maple Solution - Coded
An equation for a plane tangent to the surface z=fx,y at the point a,b,fa,b is given in position-vector form by the TangentPlane command in the Student VectorCalculus package.
This form of the plane, essentially a parametric representation, can be interpreted as the equation
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