Page:Elements of the Differential and Integral Calculus - Granville - Revised.djvu/227



 Consider the function $$f(x, y)$$. Changing $$x$$ into $$x + \delta x$$ and keeping $$y$$ constant, we get from the Theorem of Mean Value, (46), &sect; 106,

If we now change $$y$$ to $$y + \delta y$$ and keep $$x$$ and $$\delta x$$ constant, the total increment of the left-hand member of (A) is

The total increment of the right-hand member of (A) found by the Theorem of Mean Value, (46), &sect; 106, is

Since the increments (B) and (C) must be equal,

In the same manner, if we take the increments in the reverse order,

$$\theta_3$$ and $$\theta_4$$ also lying between zero and unity.

The left-hand members of (D) and (E) being identical, we have

Taking the limit of both sides as $$\delta x$$ and $$\delta y$$ approach zero as limits, we have

since these functions are assumed continuous. Placing

(G) may be written

That is, the operations of differentiating with respect to $$x$$ and with respect to $$y$$ are commutative.