Advanced Calculus using Mathematica

Advanced Calculus using Mathematica: NoteBook Edition is a complete text on calculus of several variables written in Mathematica NoteBooks.

Parametric 2D: Procedure 15.3.1

Here is the general procedure for finding the tangent to a parametric surface:

Procedure  15.3.1: Finding a Parametric Tangent Plane

Suppose AdvCalcMathWeb_150.png is smooth in a neighborhood of a particular fixed (s,t) and AdvCalcMathWeb_151.png.    To find the equation of the plane tangent to the parametric surface X[s,t] at (s,t)  do the following steps:

1.  Compute the general symbolic derivatives  AdvCalcMathWeb_152.png, AdvCalcMathWeb_153.png, AdvCalcMathWeb_154.png, AdvCalcMathWeb_155.png, AdvCalcMathWeb_156.png, AdvCalcMathWeb_157.png and total differential:

AdvCalcMathWeb_158.png  or     AdvCalcMathWeb_159.png

(The derivatives are functions of  (s,t).)

2.  Calculate the specific values of the derivatives at the particular  (s,t),  AdvCalcMathWeb_160.png, AdvCalcMathWeb_161.png, AdvCalcMathWeb_162.png, AdvCalcMathWeb_163.png, AdvCalcMathWeb_164.png, AdvCalcMathWeb_165.png, and write the specific total differential at this point:

dX = U ds+V dt  or     AdvCalcMathWeb_166.png

This is the equation of the parametric plane through the dX-origin with directions U and V and parameters  ds and dt.

3. (Optional)  If you want the equation of the tangent plane in  (x,y,z)-coordinates, say AdvCalcMathWeb_167.png, AdvCalcMathWeb_168.png, AdvCalcMathWeb_169.png is your particular point of tangency and AdvCalcMathWeb_170.png, AdvCalcMathWeb_171.png, AdvCalcMathWeb_172.png and  AdvCalcMathWeb_173.png, AdvCalcMathWeb_174.png, AdvCalcMathWeb_175.png are your two particular tangent vectors. The local  (dx,dy,dz)-coordinates are related to (x,y,z)-coordinates by AdvCalcMathWeb_176.png, AdvCalcMathWeb_177.png, and AdvCalcMathWeb_178.png, so replace  dx,  dy, and dz  with AdvCalcMathWeb_179.png, AdvCalcMathWeb_180.png,  and AdvCalcMathWeb_181.png, obtaining the equation

AdvCalcMathWeb_182.png  or  AdvCalcMathWeb_183.png  or AdvCalcMathWeb_184.png

This is the parametric plane through  AdvCalcMathWeb_185.png  with directions  AdvCalcMathWeb_186.png  and  AdvCalcMathWeb_187.png  and parameters  ds and dt.  Notice that to correctly evaluate this final expression, you need to perform steps (1), (2), and (3) in that order.

You can plot the tangent plane using the equation in local coordinates  (dx,dy,dz)  centered on the surface at the particular  X[s,t].  The graph of this equation is a parametric plane through the (dx,dy,dz)-origin centered at the point  X[s,t].


Figure 15.3.5: The Plane Tangent to a Parametric Surface

The procedure breaks down if  U  and  V are scalar multiples because the parametric graph of  AdvCalcMathWeb_189.png  is not a plane, but rather lies on a line thru  AdvCalcMathWeb_190.png in the direction of  U,  AdvCalcMathWeb_191.png,  AdvCalcMathWeb_192.png or is just a point if both tangent vectors are zero.


Find the equation of the plane tangent to the unit sphere at  (s,t)=(π/4,π/3)

    AdvCalcMathWeb_193.png,  0<s<π, 0≤t≤2 π,  and  AdvCalcMathWeb_194.png

Step 1:

    AdvCalcMathWeb_195.png  &  AdvCalcMathWeb_196.png

and all three vectors are defined for all (s,t), so the functions are smooth.  They are non-parallel for 0<s<π by the cross product computation in Example 15.4.2, 3.


Step 2:    AdvCalcMathWeb_198.png  &  AdvCalcMathWeb_199.png

    AdvCalcMathWeb_200.png  or  

Step 3:

AdvCalcMathWeb_201.png  or  AdvCalcMathWeb_202.png

You sketch the plane in local coordinates by drawing the two tangent vectors and filling in the plane as above in Figure 15.4.6

Perpendicular ("normal") to a parametric surface

The cross product  AdvCalcMathWeb_203.png  is perpendicular to both  AdvCalcMathWeb_204.png  and  AdvCalcMathWeb_205.png, hence is perpendicular to the tangent plane and the surface.  (The perpendicular to a surface is sometimes called the “normal” vector to the surface.)  In the example of the unit sphere at  (s,t)=(π/4,π/3),  



Figure 11.11.6: The Perpendicular to the Plane Tangent to a Parametric Surface