Now this is a deep concept, and we'll illustrate this with an example so

that you can gain some intuition about this.

The mathematical method of performing this change of variables

is called the Legendre transform.

Now let's consider the following steps.

Let's take a function y given by f(x) which is defined as a list of x and

y pairs, that is the plot of y of x graphs (x1, y1),

(x2, y2), (x3, y3), and so on.

Now, equivalently, it's possible to express the same information in terms

of the tangent slope c of x on the corresponding y-intercepts b,

that is the points (c1, b1), (c2, b2), (c3, b3), and so on.

Now for a small change dx in the x variable,

the corresponding change dy is given by.

Now from this point on the curve, we can extend a line back to

x equals 0 to find the y-intercept b of x such that.

Now the resulting function for

the series of intercepts versus the slope, that is b as a function of c,

that is b of c, contains the same information as y equals y of x.

Now let's say we're interested in a change in the intercept b of c.

Now, this is given by.