In the applications of the Calculus, especially in geometry, it is usually most convenient to work with equations expressed not, like equation (1) of § 154, in terms of the increments $$\delta x$$, $$\delta y$$, $$\delta z$$ of the functions $$x$$, $$y$$, $$z$$, but in terms of what are called their differentials $$dx$$, $$dy$$, $$dz$$.

Let us return for a moment to a function $$y = f(x)$$ of a single variable $$x$$. If $$f'(x)$$ is continuous then $\begin{equation*} \delta y = \{f'(x) + \epsilon\}\, \delta x, \tag{1} \end{equation*}$ where $$\epsilon \to 0$$ as $$\delta x \to 0$$: in other words the equation $\begin{equation*} \delta y = f'(x)\, \delta x \tag{2} \end{equation*}$ is ‘approximately’ true. We have up to the present attributed no meaning of any kind to the symbol $$dy$$ standing by itself. We now agree to define $$dy$$ by the equation $\begin{equation*} dy = f'(x)\, \delta x. \tag{3} \end{equation*}$

If we choose for $$y$$ the particular function $$x$$, we obtain $\begin{equation*} dx = \delta x, \tag{4} \end{equation*}$ so that $\begin{equation*} dy = f'(x)\, dx. \tag{5} \end{equation*}$ If we divide both sides of (5) by $$dx$$ we obtain $\begin{equation*} \frac{dy}{dx} = f'(x), \tag{6} \end{equation*}$ where $$dy/dx$$ denotes not, as heretofore, the differential coefficient of $$y$$, but the quotient of the differentials $$dy$$, $$dx$$. The symbol $$dy/dx$$ thus acquires a double meaning; but there is no inconvenience in this, since (6) is true whichever meaning we choose.

The equation (5) has two apparent advantages over (2). It is exact and not merely approximate, and its truth does not depend on any assumption as to the continuity of $$f'(x)$$. On the other hand it is precisely the fact that we can, under certain conditions, pass from the exact equation (5) to the approximate equation (2), which gives the former its importance. The advantages of the ‘differential’ notation are in reality of a purely technical character. These technical advantages are however so great, especially when we come to deal with functions of several variables, that the use of the notation is almost inevitable.

When $$f'(x)$$ is continuous, we have $\lim \frac{dy}{\delta y} = 1$ when $$\delta x \to 0$$. This is sometimes expressed by saying that $$dy$$ is the principal part of $$\delta y$$ when $$\delta x$$ is small, just as we might say that $$ax$$ is the ‘principal part’ of $$ax + bx^{2}$$ when $$x$$ is small.

We pass now to the corresponding definitions connected with a function $$z$$ of two independent variables $$x$$ and $$y$$. We define the differential $$dz$$ by the equation $\begin{equation*} dz = f_{x}’\, \delta x + f_{y}’\, \delta y. \tag{7} \end{equation*}$ Putting $$z = x$$ and $$z = y$$ in turn, we obtain $\begin{equation*} dx = \delta x,\quad dy = \delta y, \tag{8} \end{equation*}$ so that $\begin{equation*} dz = f_{x}’\, dx + f_{y}’\, dy, \tag{9} \end{equation*}$ which is the exact equation corresponding to the approximate equation (1) of § 154. Here again it is to be observed that the former is of importance only for reasons of practical convenience in working and because the latter can in certain circumstances be deduced from it.

One property of the equation (9) deserves special remark. We saw in § 153 that if $$z = f(x, y)$$, $$x$$ and $$y$$ being not independent but functions of a single variable $$t$$, so that $$z$$ is also a function of $$t$$ alone, then $\frac{dz}{dt} = \frac{\partial f}{\partial x}\, \frac{dx}{dt} + \frac{\partial f}{\partial y}\, \frac{dy}{dt}.$ Multiplying this equation by $$dt$$ and observing that $dx = \frac{dx}{dt}\, dt,\quad dy = \frac{dy}{dt}\, dt,\quad dz = \frac{dz}{dt}\, dt,$ we obtain $dz = f_{x}’\, dx + f_{y}’\, dy,$ which is the same in form as (9). Thus the formula which expresses $$dz$$ in terms of $$dx$$ and $$dy$$ is the same whether the variables $$x$$ and $$y$$ are independent or not. This remark is of great importance in applications.

It should also be observed that if $$z$$ is a function of the two independent variables $$x$$ and $$y$$, and $dz = \lambda\, dx + \mu\, dy,$ then $$\lambda = f_{x}’$$, $$\mu = f_{y}’$$. This follows at once from the last paragraph of § 154.

It is obvious that the theorems and definitions of the last three sections are capable of immediate extension to functions of any number of variables.

Example LXII

1. The area of an ellipse is given by $$A = \pi ab$$, where $$a$$, $$b$$ are the semiaxes. Prove that $\frac{dA}{A} = \frac{da}{a} + \frac{db}{b},$ and state the corresponding approximate equation connecting the increments of the axes and the area.

2. Express $$\Delta$$, the area of a triangle $$ABC$$, as a function of (i) $$a$$, $$B$$, $$C$$, (ii) $$A$$, $$b$$, $$c$$, and (iii) $$a$$, $$b$$, $$c$$, and establish the formulae $\begin{gathered} \frac{d\Delta}{\Delta} = 2\frac{da}{a} + \frac{c\, dB}{a\sin B} + \frac{b\, dC}{a\sin C},\quad \frac{d\Delta}{\Delta} = \cot A\, dA + \frac{db}{b} + \frac{dc}{c},\\ d\Delta = R(\cos A\, da + \cos B\, db + \cos C\, dc),\end{gathered}$ where $$R$$ is the radius of the circumcircle.

3. The sides of a triangle vary in such a way that the area remains constant, so that $$a$$ may be regarded as a function of $$b$$ and $$c$$. Prove that $\frac{\partial a}{\partial b} = -\frac{\cos B}{\cos A},\quad \frac{\partial a}{\partial c} = -\frac{\cos C}{\cos A}.$

[This follows from the equations $da = \frac{\partial a}{\partial b}\, db + \frac{\partial a}{\partial c}\, dc,\quad \cos A\, da + \cos B\, db + \cos C\, dc = 0.]$

4. If $$a$$, $$b$$, $$c$$ vary so that $$R$$ remains constant, then $\frac{da}{\cos A} + \frac{db}{\cos B} + \frac{dc}{\cos C} = 0,$ and so $\frac{\partial a}{\partial b} = -\frac{\cos A}{\cos B},\quad \frac{\partial a}{\partial c} = -\frac{\cos A}{\cos C}.$

[Use the formulae $$a = 2R\sin A$$, …, and the facts that $$R$$ and $$A + B + C$$ are constant.]

5. If $$z$$ is a function of $$u$$ and $$v$$, which are functions of $$x$$ and $$y$$, then $\frac{\partial z}{\partial x} = \frac{\partial z}{\partial u}\, \frac{\partial u}{\partial x} + \frac{\partial z}{\partial v}\, \frac{\partial v}{\partial x},\quad \frac{\partial z}{\partial y} = \frac{\partial z}{\partial u}\, \frac{\partial u}{\partial y} + \frac{\partial z}{\partial v}\, \frac{\partial v}{\partial y}.$

[We have $dz = \frac{\partial z}{\partial u}\, du + \frac{\partial z}{\partial v}\, dv,\quad du = \frac{\partial u}{\partial x}\, dx + \frac{\partial u}{\partial y}\, dy,\quad dv = \frac{\partial v}{\partial x}\, dx + \frac{\partial v}{\partial y}\, dy.$ Substitute for $$du$$ and $$dv$$ in the first equation and compare the result with the equation $dz = \frac{\partial z}{\partial x}\, dx + \frac{\partial z}{\partial y}\, dy.]$

6. Let $$z$$ be a function of $$x$$ and $$y$$, and let $$X$$, $$Y$$, $$Z$$ be defined by the equations $x = a_{1} X + b_{1} Y + c_{1} Z,\quad y = a_{2} X + b_{2} Y + c_{2} Z,\quad z = a_{3} X + b_{3} Y + c_{3} Z.$ Then $$Z$$ may be expressed as a function of $$X$$ and $$Y$$. Express $$\partial Z/\partial X$$, $$\partial Z/\partial Y$$ in terms of $$\partial z/\partial x$$, $$\partial z/\partial y$$. [Let these differential coefficients be denoted by $$P$$, $$Q$$ and $$p$$, $$q$$. Then $$dz – p\, dx – q\, dy = 0$$, or $(c_{1} p + c_{2} q – c_{3})\, dZ + (a_{1} p + a_{2} q – a_{3})\, dX + (b_{1} p + b_{2} q – b_{3})\, dY = 0.$ Comparing this equation with $$dZ – P\, dX – Q\, dY = 0$$ we see that $P = -\frac{a_{1}p + a_{2}q – a_{3}}{c_{1}p + c_{2}q – c_{3}},\quad Q = -\frac{b_{1}p + b_{2}q – b_{3}}{c_{1}p + c_{2}q – c_{3}}.]$

7. If $(a_{1} x + b_{1} y + c_{1} z)p + (a_{2} x + b_{2} y + c_{2} z)q = a_{3} x + b_{3} y + c_{3} z,$ then $(a_{1} X + b_{1} Y + c_{1} Z) P + (a_{2} X + b_{2} Y + c_{2} Z) Q = a_{3} X + b_{3} Y + c_{3} Z.$

8. Differentiation of implicit functions. Suppose that $$f(x, y)$$ and its derivative $$f_{y}'(x, y)$$ are continuous in the neighbourhood of the point $$(a, b)$$, and that $f(a, b) = 0,\quad f_{b}'(a, b) \neq 0.$ Then we can find a neighbourhood of $$(a, b)$$ throughout which $$f_{y}'(x, y)$$ has always the same sign. Let us suppose, for example, that $$f_{y}'(x, y)$$ is positive near $$(a, b)$$. Then $$f(x, y)$$ is, for any value of $$x$$ sufficiently near to $$a$$, and for values of $$y$$ sufficiently near to $$b$$, an increasing function of $$y$$ in the stricter sense of § 95. It follows, by the theorem of § 108, that there is a unique continuous function $$y$$ which is equal to $$b$$ when $$x = a$$ and which satisfies the equation $$f(x, y) = 0$$ for all values of $$x$$ sufficiently near to $$a$$.

Let us now suppose that $$f(x, y)$$ possesses a derivative $$f_{x}'(x, y)$$ which is also continuous near $$(a, b)$$. If $$f(x, y) = 0$$, $$x = a + h$$, $$y = b + k$$, we have $0 = f(x, y) – f(a, b) = (f_{a}’ + \epsilon) h + (f_{b}’ + \eta) k,$ where $${\epsilon}$$ and $$\eta$$ tend to zero with $$h$$ and $$k$$. Thus $\frac{k}{h} = -\frac{f_{a}’ + \epsilon}{f_{b}’ + \eta} \to -\frac{f_{a}’}{f_{b}’},$ or $\frac{dy}{dx} = -\frac{f_{a}’}{f_{b}’}.$

9. The equation of the tangent to the curve $$f(x, y) = 0$$, at the point $$x_{0}$$, $$y_{0}$$, is $(x – x_{0}) f_{x_{0}}'(x_{0}, y_{0}) + (y – y_{0}) f_{y_{0}}'(x_{0}, y_{0}) = 0.$