Page:Calculus Made Easy.pdf/196

 variables, is to write the differential coefficients with Greek deltas, like $$\partial$$, instead of little $$d$$. In this way

If we put in these values for $$v$$ and $$u$$ respectively, we shall have {{c|$$ \left. \begin{aligned} dy_v = \frac{\partial y}{\partial u}\, du, \\ dy_u = \frac{\partial y}{\partial v}\, dv, \end{aligned} \right\}\; $$which are partial differentials.}} But, if you think of it, you will observe that the total variation of $$y$$ depends on both these things at the same time. That is to say, if both are varying, the real $$dy$$ ought to be written

and this is called a total differential. In some books it is written $$dy = \left(\dfrac{dy}{du}\right)\, du + \left(\dfrac{dy}{dv}\right)\, dv$$.

Example (1). Find the partial differential coefficients of the expression $$w = 2ax^2 + 3bxy + 4cy^3$$. The answers are: {{c|$$ \left. \begin{aligned} \frac{\partial w}{\partial x} &= 4ax + 3by. \\ \frac{\partial w}{\partial y} &= 3bx + 12cy^2. \end{aligned} \right\} $$}}