Implicit Curves, Surfaces, and Contour Plots

Contour plots, similar to the topographical maps used by mountain hikers, are an important way to study functions of several variables.  These plots correspond to a family of curves of constant height as shown in the next figure.

Contours as Height Changes

Click the link below this figure to run an animation that shows the various level curves as the height of the horizontal slicing plane changes.

Load the Animation 

The whole collection of these curves can be graphed by the following command.

We will also study implicit graphs in 3D as in the next example.

Example

Find the equation tangent to  =1  at  (x,y,z) = (5/2,1,)

First, the general symbolic total differential of the equation is

At the particular point  (x,y,z) = (5/2,1,), the differential is

        ⟺     = 0

The tangent is the plane through  (5/2,1,)  perpendicular to  P = (6,15,) ≈ (6,15,14.14) as shown next with the implicit ellipsoid.