Total Differential

    Recall, from the calculus of one variable: if where is a differentiable function, then differential represents the amount that the tangent line rises or falls, whereas represents the amount that the curve rises or falls when changes by an amount Since we have when is small. If we take then we have which says that the actual change in is approximately equal to the differential If is a known number and it is desired to calculate an approximate value for where is small, then



yields the approximation,

.

We will use the total differential to do the same for functions of two or more variables.

Definition (Total Differential) For a function of two variables , if and are given increments and , then the corresponding increment of is



The differentials and are independent variables; that is, they can be given any values. Then the differential , also called the total differential, is defined by

Example (Total Differential) If , find the differential Further, if changes from to and changes from to , compare the values of and Which is easier to compute or ?

    Solution. By definition,



Putting , , , and , we get



The increment of is







Notice that but is easier to compute.

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Total Differential
Published by Library of Math -- Online math organized by subject into topics.
Written by Smith, David A.
http://www.libraryofmath.com/total-differential.html