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176

Calculus Made Easy

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

${\begin{aligned}{\frac {\partial y}{\partial u}}&=v,\\{\frac {\partial y}{\partial v}}&=u.\end{aligned}}$

If we put in these values for $v$ and $u$ respectively, we shall have

$\left.{\begin{aligned}dy_{v}={\frac {\partial y}{\partial u}}\,du,\\dy_{u}={\frac {\partial y}{\partial v}}\,dv,\end{aligned}}\right\}\;$ 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

$dy={\frac {\partial y}{\partial u}}\,du+{\dfrac {\partial y}{\partial v}}\,dv$ ;

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:

$\left.{\begin{aligned}{\frac {\partial w}{\partial x}}&=4ax+3by.\\{\frac {\partial w}{\partial y}}&=3bx+12cy^{2}.\end{aligned}}\right\}$

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