ABSTRACT
The variation of the dependent variable y with the independent variable x can be represented as the locus of points satisfying the relation y y(x), as shown in Fig. C.1, or as the envelope of a family of tangent lines shown in Fig. C.2. In Figure C.1 every point in the plane is described by two numbers x and y and every line in the plane in Fig. C.2 can be described by two numbers m and , where m is the slope of the line and is its intercept with the y-axis. Then, just as the relation y y(x) selects a subset of all possible points (x,y), a relation (m) selects a subset of all possible lines (m,). Knowledge of the intercept of the tangent lines as a function of the slopes m allows the construction of the family of tangent lines and thus the curve for which they are the envelope. Thus the relationship
(C.1) is equivalent to the relation y y(x), and in Eq. (C.1) m is the independent variable. The computation of the relation (m) from the known relation y y(x) is known as a Legendre transformation. Fig. C.3 shows a tangent line of slope m and going through the point x,y. If the intercept is then
(C.2)
or
(C.3) Differentiation of the known equation y y(x) gives m m(x) and elimination of x and y gives the desired relation and m. The function is known as the Legendre
y mx
m y x 0
(m)
Figure C.1 The locus of points satisfying the relation y y (x).
