MAT37x

Franklin College

Spring2004

Erich Prisner,

The Cassini surface is a function from
**R ^{2}**
into

We will always consider only the function f with parameter b=2, thus we get

Here are the partial derivatives:

Obviously these expressions are not defined at the two critical points (-2,0) and (2,0).
The level lines of these functions
f_{x}
and
f_{y}
are shown below.

Remember that level lines cannot cross. Therefore what looks like a crossing of level lines at
(-2,0)
and (2,0)
only means that these level lines come closer and closer when approaching the two points
(where the functions
f_{x}
and
f_{y}
are not defined).

Note that the level line of
f_{x}
of level 0
are all points where the level lines of the original Cassini surface
have a horizontal tangent. In our example it is the circle with center
(0,0)
and radius 2
. In the same way, the level line of
f_{y}
of level 0
are all points where the level lines of the original Cassini surface have
a vertical tangent. In our example it is the x
-axis.

The level lines of the Cassini surface are the so-called Cassini curves. Look at the Cassini curve of level h , recall it has the equation:

Using implicit differentiation (use product rule, and remember that y=y(x) depends on x) with respect to x, we obtain

Then we solve this equation for dy/dx and get

On the right is the direction field of this first order differential equation. Of course, the solutions are just the Cassini curves---the level lines of the surface.

Remember that another way to obtain dy/dx uses the partial derivatives and the formula
dy/dx = -f_{x}/f_{y}.
Try it to see that we get the same result.

The gradient has both direction and absolute value. The direction is depicted to the right, it is perpendicular to the direction field shown above. Again we have the connection to differential equations.

What is the absolute value of the gradient vector
(f_{x},f_{y})? It is the square root of
f_{x}^{2}+f_{y}^{2}.
If we substitute the values above and simplify, we get the answer

So the surprising answer is that the steepness of the Cassini surface only depends on the distance of the point to (0,0), it is twice this distance. Therefore there can only be one saddle point---the point (0,0) itself, and there are no local maximum or minimum points, at least not where the function is differentiable. Of course there are two minimum points at nondifferentiable (critical) points (2,0) and (-2,0).