Some notes on Chapter 1 by Professor Goodman

Note 1.  Explicit, Implicit and Parametric

The subject of sections 1-3 of the text is three different types of mathematical descriptions or specifications  of one type of mathematical object.  

The one type of object is a "graph".  In two dimensions, a graph is a curve in the  two-dimensional plane. In three dimensions, a graph is a surface in three dimensional space.

The three different types of  descriptions or specifications are called "explicit specification", "implicit specification" (or level curve specification) , and"parametric specification".   

Please carefully compare the following definitions with the ones given in the text.

Definition 1.1.: An explicit function and its graph

Definition 1.2.1:  A relation defines a function implicitly, and determines a curve as a level set.

Comments:  
1.  The dependence of y on x is specified implicitly by the relation F[x, y] = k.   

2.  On the other hand, the curve  {(x, y)  :  F[x, y] = k} is specified as a level set of the function  F[x,y].  In some sense, it is also fair to say that the curve is specified implicitly by the relation F[x,y]=k , as it can be difficult to actually compute points on the curve.

Definition 1.3.1: Parametric equations and graph

A circle

The unit circle in 2-D centered at the origin is the set of points in the plane at distance 1 from the origin.   The circle is neither explicit, implicit or parametric.  It is just a circle.

However it can be specified by explicit functions, or as a level set,  or parametrically, as follows

The circle can be specified as a level set by the equation:

This same equation implicitly defines y as a function of x.

The circle can be specified parametrically by the pair of equations

Finally (and perhaps least conveniently)  the circle can be specified in two pieces by explicit functions.  The top half of the circle is given explicitly as

while the bottom half of the circle is given explicitly as

A more complicated level curve

Let us consider the  relation

This equation defines some curve as a level curve of the function

Here is a plot of the curve in a region near the origin (0,0)

Clear[x, y, F, a, b, c, d, k];
F[x_, y_] =  
        (x^3) -  3 x +  5 y - 3 (y^3) + (y^2);
k = 4;
a = -5;
b  =  5;
c = -5;
d  =  5;
ContourPlot[F[x, y], {x, a, b}, {y, c, d}, Contours -> {k}, 
      ContourShading -> False]; F[x, y] == k

We don't know how to explicitly solve for y in terms of x or to solve for x in terms of y, but we can see that the curve can be split up into pieces such that each piece is the graph of a function -- either y is a function of x, or x is a function of y.

Summary comments:  
The curve or graph itself is not explicit, implicit, or parametric.  It is just a curve.  
Only the specification of the graph can be explicit, implicit (curve specified as a level set) or parametric. So, when Professor Stroyan writes in the text that a certain graph is an "explicit graph", we will read that as saying that the graph is specified by an explicit function.  When Professor Stroyan writes that a graph is an "implicit graph", we will read that as saying that the graph is specified as a level set of some function of two variables.  When Professor Stroyan write that a graph is a "parametric graph", we will read that as saying that the graph is specified parametrically.

Note 2.  Really Explicit?

When we say that a function  is given explicitly as y=f[x], we do not necessarily mean that an explicit formula is known for the function.  

Here is a perfectly good physically defined function:  f[x] is the temperature, in degrees Celsius in my back yard during the next 24 hours.  

Here is the rule for computing the function:  to find the value of the function at time x, march into my backyard with a thermometer at time x and read the thermometer.  Or, better,  leave a recording thermometer in my backyard and come back in 24 hours (because it's really to cold to be out there).

In our terminology, the graph of  y = f[x] counts as an explicitly specified graph, even though we don't have an explicit formula for the function.

Note 3.  How to compute the tangent line to an implicitly defined graph

Consider an relation  F[x, y]  =  k, defining a curve in the plane.  
We will take as our example the relation (x^2) + (y^2) = 1, so  F[x, y] = (x^2) + (y^2) and k  =  1.
If ((Subscript[x, 0,]) (Subscript[y, 0])) is some point on the relation, i.e. F[(Subscript[x, 0]), (Subscript[y, 0])]  =  k, we want to compute the tangent line to the curve at ((Subscript[x, 0,]) (Subscript[y, 0])).

The rule for doing this involves partial derivatives of the function F[x, y].
The Mathematica notation for a partial derivative is (∂_x) F[x, y], while the "blackboard" notation is ∂(F/∂)x[x, y].
In either notation the partial derivative with respect to x  is just the derivative with respect to x, treating y as a constant.
So for  F[x, y] = (x^2) + (y^2), we have (∂_x) F[x, y]  =  2  x.  
The partial derivative with respect to y is just the derivative with respect to y, treating x as a constant.
So for  F[x, y] = (x^2) + (y^2), we have (∂_y) F[x, y]  =  2  y.

OK, now we are ready for the FACT about partial derivatives and the tangent line.
Evaluate the two partial derivatives at the point ((Subscript[x, 0,]) (Subscript[y, 0])), say

Then (A, B) is the direction perpendicular to the tangent line to the curve at ((Subscript[x, 0,]) (Subscript[y, 0])).  Therefore the equation of the tangent line is

And what is the "something".   Since ((Subscript[x, 0,]) (Subscript[y, 0])) is a point on the tangent line, "something" must be

Thus the equation of the tangent line is

OK, let's do a completely concrete example.  Our  F[x, y]  is  F[x, y] = (x^2) + (y^2).  Our relation is F[x, y] = (x^2) + (y^2)  =  1.
A point on the relation is ((Subscript[x, 0,]) (Subscript[y, 0]))  =  (1/2,  (Sqrt[3])/
            2).   The partial derivatives evaluated at this point are

Thus the  equation of the line is

That is,

Find the tangent line to (x^2) + (y^2) = 3 at the point (1,  (Sqrt[2])), using the procedure explained above.  Then solve for y as an explicit function of x, and compute the tangent line again using the derivative of the explicit function. Compare the results.  Draw the graph of the relation and the tangent line at the indicated point.

Find the tangent line to (x^2) -  (y^2) = 3 at the point (3,  (Sqrt[6])), using the procedure explained above.  Then solve for y as an explicit function of x, and compute the tangent line again using the derivative of the explicit function. Compare the results.  Draw the graph of the relation and the tangent line at the indicated point.

Consider the parametrically specified curve

Find the tangent line to the curve at the point corresponding to t  =  π/4;  it is best to write the tangent line in parametric form.