TOTAL DIFFERENTIALS

3. TOTAL DIFFERENTIALS

The graph (Figure 3.1) of the equation y = f (x) is a curve in the (x, y) plane and

is the slope of the tangent to the curve at the point (x, y). In calculus, we use ∆x to mean a change in x, and ∆y means the corresponding change in y (see Figure 3.1). By definition

∆x→0 ∆x

Figure 3.1

We shall now define the differential dx of the independent variable as (3.3)

dx = ∆x.

However, dy is not the same as ∆y. From Figure 3.1 and equation (3.1), we can see that ∆y is the change in y along the curve, but dy = y ′ dx is the change in y along the tangent line. We say that dy is the tangent approximation (or linear approximation) to ∆y.

Example. If y = f (t) represents the distance a particle has gone as a function of t, then dy/dt is the speed. The actual distance the particle has gone between time t and time t + dt is ∆y. The tangent approximation dy = (dy/dt)dt is the distance it would have gone if it had continued with the same speed dy/dt which it had at time t.

You can see from the graph (Figure 3.1) that dy is a good approximation to ∆y if dx is small. We can say this more exactly using (3.2). Saying that dy/dx is the limit of ∆y/∆x as ∆x → 0 means that the difference ∆y/∆x − dy/dx → 0 as ∆x → 0. Let us call this difference ǫ; then we can say

where ǫ → 0 as ∆x → 0, or since dx = ∆x (3.5)

∆y = (y ′ + ǫ)dx, where ǫ → 0 as ∆x → 0.

194 Partial Differentiation Chapter 4

The differential dy = y ′ dx is called the principal part of ∆y; since ǫ is small for small dx, you can see from (3.5) that dy is then a good approximation to ∆y.

In our example, suppose y = t 2 , t = 1, dt = 0.1. Then ∆y = (1.1) 2 −1 2 = 0.21,

dy dy =

dt = 2 · 1 · (0.1) = 0.2, dt

∆y = (y ′ + ǫ)dt = (2 + 0.1)(0.1) = dy + ǫdt = 0.2 + 0.01.

Thus dy is a good approximation to ∆y. For a function of two variables, z = f (x, y), we want to do something similar to

this. We have said that this equation represents a surface and that the derivatives ∂f /∂x, ∂f /∂y, at a point, are the slopes of the two tangent lines to the surface in the x and y directions at that point. The symbols ∆x = dx and ∆y = dy represent changes in the independent variables x and y. The quantity ∆z means the corresponding change in z along the surface. We define dz by the equation

The differential dz is called the total differential of z. Let us consider the geometrical meaning of dz. Recall (Figure 3.1) that for y = f (x), dy was the change in y along the tangent line; here we shall see that dz is the change in z along the tangent plane. In Figure 3.2, P QRS is a surface, P ABC is the plane tangent to the surface at P , and P DEF is a horizontal plane through P . Thus P SCF is the plane y = const. (through P ), P S is the curve of intersection of this plane with the surface, and P C is the tangent line to this curve and so has the slope ∂f /∂x; then (just as in Figure 3.1), if P F = dx, we have CF = (∂f /∂x) dx. Similarly, P QAD is a plane x = const., intersecting the surface in the curve P Q, whose tangent is P A; with

Section 3 Total Differentials 195

P D = dy, we have DA = (∂f /∂y) dy. From the figure, GE = CF , and BG = AD, so

Thus, as we said, dz is the change in z along the tangent plane when x changes by dx and y by dy. In the figure, ER = ∆z, the change in z along the surface.

From the geometry, we can reasonably expect dz to be a good approximation to ∆z if dx and dy are small. However, we should like to say this more accurately in an equation corresponding to (3.5). We can do this if ∂f /∂x and ∂f /∂y are continuous functions. By definition

∆z = f (x + ∆x, y + ∆y) − f(x, y).

By adding and subtracting a term, we get (3.8)

∆z = f (x + ∆x, y) − f(x, y) + f(x + ∆x, y + ∆y) − f(x + ∆x, y). Recall from calculus that the mean value theorem (law of the mean) says that for

a differentiable function f (x), (3.9)

f (x + ∆x) − f(x) = (∆x)f ′ (x 1 ),

where x 1 is between x and x + ∆x. Geometrically this says (Figure 3.3) that there is a tangent line somewhere between x and x + ∆x which has the same slope as the

Figure 3.3

line AB. In the first two terms of the right side of (3.8), y is constant, and we can use (3.9) if we write ∂f /∂x for f ′ . In the last two terms of (3.8), x is constant and

we can use an equation like (3.9) with y as the variable; y 1 will mean a value of y between y and y + ∆y. Then (3.8) becomes

If the partial derivatives of f are continuous, then their values in (3.10) at points near (x, y) differ from their values at (x, y) by quantities which approach zero as

∆x and ∆y approach zero. Let us call these quantities ǫ 1 and ǫ 2 . Then we can write

(ǫ 1 and ǫ 2 → 0 as ∆x and ∆y → 0), where ∂f /∂x and ∂f /∂y in (3.11) are evaluated at (x, y). Equation (3.11) [like (3.5)

for the y = f (x) case] tells us algebraically what we suspected from the geometry,

196 Partial Differentiation Chapter 4

that (if ∂f /∂x and ∂f /∂y are continuous ) dz is a good approximation to ∆z for small dx and dy. The differential dz is called the principal part of ∆z.

Everything we have said about functions of two variables works just as well for functions of any number of variables. if u = f (x, y, z, · · · ), then by definition

and du is a good approximation to ∆u if the partial derivatives of f are continuous and dx, dy, dz, etc., are small.

PROBLEMS, SECTION 3

1. Consider a function f (x, y) which can be expanded in a two-variable power series, (2.3) or (2.7). Let x − a = h = ∆x, y − b = k = ∆y; then x = a + ∆x, y = b + ∆y so that f (x, y) becomes f (a + ∆x, b + ∆y). The change ∆z in z = f (x, y) when x changes from a to a + ∆x and y changes from b to b + ∆y is then

∆z = f (a + ∆x, b + ∆y) − f(a, b). Use the series (2.7) to obtain (3.11) and to see explicitly what ǫ 1 and ǫ 2 are and

that they approach zero as ∆x and ∆y → 0.