Differential Calculus

Cover Differential Calculus
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If z is kept constant, u is again a function of x and y. Its partial derivative appears on the right side of the equation. We must not of course use the same symbol for both of these derivatives. A way to avoid the confusion is to use the Chap. XI. PARTIAL DIFFERENTIATION 127 letter/ instead of u on the right side of the equation. It then becomes dx dx dz dx It is understood that / {x, y, z) is a definite function of x, y, z and that ^ is the derivative obtained with all the variables dx but X ...constant.
97. Change of Variable. — If w is a function of x and y we have said that the equation , 5m , , du I du == —- dx + ^- ay dx dy is true whether x and y are the independent variables or not.
To show this let s and t be the independent variables and x and y functions of them. Then, by definition, , du , , du ,, du = -r- as + -rr dt.
dS Of Since m is a function of x and j/" which are functions of s and t, by equation (96a), du _ du dx du dy du _ du dx du dy ds ~ dx ds dy ds ' Tt ^ dx 'di dy 'di\ Consequently, , /du dx , du dy\ , , /du dx , du dy\ ,.


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