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 is smooth in a neighborhood of a particular fixed (s,t) and .    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  , , , , , and total differential:

or

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

2.  Calculate the specific values of the derivatives at the particular  (s,t),  , , , , , , and write the specific total differential at this point:

dX = U ds+V dt  or

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 , , is your particular point of tangency and , , and  , , are your two particular tangent vectors. The local  (dx,dy,dz)-coordinates are related to (x,y,z)-coordinates by , , and , so replace  dx,  dy, and dz  with , ,  and , obtaining the equation

or    or

This is the parametric plane through    with directions    and    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    is not a plane, but rather lies on a line thru   in the direction of  U,  ,   or is just a point if both tangent vectors are zero.

Example

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

,  0<s<π, 0≤t≤2 π,  and

Step 1:

&

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:      &

or

Step 3:

or

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    is perpendicular to both    and  , 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