The definition of parametric representation is that a surface can be described as images of vector-valued functions. The most common example of a parametric representation in 2 space involves the circle defined as a graph by :

When using a parametric surface, we define the tangent plane to be the plane passing through f(u0,v0) and parallel to the partial derivatives with respect to both u and v. The plane is given parametrically by:

**t(s,t) = s(partial w/ respect to u at (u0,v0)) +
t(partial w/ respect to v at
(u0,v0)) + f(u0,v0)**

f(u,v) = (ucos v, usin v, v), where u=0..2, and v=0..2*Pi at (u,v) = (1, 2*Pi)

partial of u = (cos v, sinv, 0) = (1, 0, 0)

partial of v = (-usinv, ucos v, 1) = (0, 1, 1)

t(s,t) = s(1, 0, 0) + t(0, 1, 1) + (1, 0, 2*Pi) = (s+1, t, t+2*Pi) for s=-infinity..infinity, t=-infinity..infinity

**NOTE: Use the variables s and t in the Maple ParamTan command**
(ParamTan will not recognize u and v)

**A.** Find the partial derivatives with
respect to s and t, and
evaluate them at (s0,t0)=(1, 1), our point of
interest.

**B.** These two vectors are the basis vectors
for your plane.
Turn them into a plane. **Hint:** this is a
vector valued function.

**C.** Sketch your surface and tangent plane
around your point
(s0,t0)=(1, 1). You can use maple if you
really want to, we allow it. : )

**D.** Check your answer using the command
ParamTan. (If you have not already pasted in the maple code that auxilliary
functions, then do so now.) The syntax is as follows:

ParamTan (function defined as 3 vector, s=smin..smax, t=tmin..tmax,point=[s0,t0]);

**Note:** These are the actual steps that *Maple* is going through when it calculates the plane.

Next:

Back: **tangent planes to
graphs**