By A. F. Bermant

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**Additional resources for A Course of Mathematical Analysis, Part II **

**Example text**

The absolute value of the partial derivative az/ax = I~(x, y) or azjay = I~ (x, y) gives the rate of change ofthe function z = I(x, y) when the argument P (x, y) moves along the straight line y = const or x = const, whilst the sign of the partial derivative I~ or I~ indicates the type of change (increase, decrease). e. x (Fig. 14a). It is clear from the figure that in the present case I~ (xo' Yo) < o. e. f~ (xo, Yo) = tan {3 (Fig. 14b). It is clear from the figure that in the present case f~(xo, Yo) > o.

All be continuous functions of their arguments. u = cp(t, v, w, ... ) = F(x, y, z, ... ). For, in view of the continuity of functions '/fl, ~, 'fj, ... , an infinitesimal displacement of the point P«x, y, z, ... ) produces infinitesimal variations of variables t, v, w, ... , which in turn produce an infinitesimal increment in the variable u by virtue of the continuity of function cpo As a result, an infinitesimal variation in u corresponds to an infinitesimal displacement of the point P (x, y, z, ...

The maximum relative error of the quotient is equal to the sum of the relative errors 01 top and bottom. 148. Directional Derivatives. 1. Let the argument of the function I(x, y)-the point P(x, y)-vary along a given radius vector, drawn from the point Po and forming an angle ex with the positive direction of Ox. We have (Sec. ; Xo t(P) = = f(x o + e cos ex, Yo + e sin (X). = f(x o + e cos ex, Yo + e sin (X) e ex ) I (*) We form the ratio f(P) - f(P o) PP o - f(x o' Yo) . (**) This is the ratio of the increment LI Z = 1(P) - 1(Po) of the function Z = f(P) as a function of the singl~ variable e to the increment of this variable (since the value e = 0 oorresponds to Po)' Now let point P tend to point P fj along the radius vector.

### A Course of Mathematical Analysis, Part II by A. F. Bermant

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