Chapter 4: Partial Differentiation
Section 4.6: Surface Normal and Tangent Plane
At P:2,−3 on the surface defined by z=fx,y≡5−x2/3−y2/2, obtain and draw both the normal and tangent plane.
Figure 4.6.1(a) shows the surface in green, the tangent plane at Q:2,−3,−5/6 in red, and the normal at this point in black.
According to Table 4.6.1, N is obtained by evaluating −fx i−fy j+k at x,y=2,−3, yielding
The tangent plane is then given vectorially by
use plots, Student:-VectorCalculus in
Figure 4.6.1(a) Surface, normal, and tangent plane
and then by
=4 x3−3 y+z−83−9+56
=4 x3−3 y+z−656
Maple Solution - Interactive
A complete solution is available with the Student MultivariateCalculus package.
Let Q be the point on the surface that corresponds to the point x,y=2,−3.
Tools≻Load Package: Student Multivariate Calculus
Context Panel: Assign Function
fx,y=5−x2/3−y2/2→assign as functionf
Context Panel: Assign Name
Obtain a surface normal at point Q
Calculus palette: Partial differentiation operator
Context Panel: Evaluate and Display Inline
Context Panel: Evaluate at a Point≻x=2,y=−3
(See Figure 4.6.1(b).)
Context Panel: Assign to a Name≻N
Figure 4.6.1(b) Dialog: Evaluate at a Point
−∂∂ x f(x,y)−∂∂ y f(x,y)1 = 23⁢xy1→evaluate at point43−31→assign to a nameN
Obtain an equation for the tangent plane
Write a sequence of names for 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
Q,N→make plane<< Plane 1 >>→representation43⁢x−3⁢y+z=656
Maple also supports a solution from first principles.
Convert Q to the position vector A
Write the name for point Q.
Context Panel: Conversions≻Column Vector
Context Panel: Assign to a Name≻A
Q = 2,−3,−56→to Vector2−3−56→assign to a nameA
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.
Obtain a vector normal to the surface
Use the diff and eval commands to obtain partial derivatives evaluated on the surface.
Use the Plane command to generate the plane data-structure.
Use the GetRepresentation command to extract the equation of the tangent plane.
The tangent plane can also be obtained via the TangentPlane command in the Student VectorCalculus package.
Student:-VectorCalculus:-TangentPlanefx,y,x=2,y=−3 = xy−43⁢x+656+3⁢y
The plane is given in the form of a position vector, where the third component is interpreted as the equation z=−43⁢x+656+3⁢y.
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