By Goursat E.
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Extra resources for A Course in Mathematical Analysis
A form, in general, a cone and In the demonstration of the general theorem on implicit functions Our geometrical that the derivative F^ did not vanish. we assumed intuition explains the necessity of this condition in general. For, is parallel to the % axis, the if but 3= tangent plane 0, F^ F^ = line parallel to the z axis and near the line x = x w y = y meets the surface, in general, in two points near the point of Hence, in general, the equation (4) would have two tangency. roots which both approach z when x and y approach x and y and a , respectively.
Which has been in use longer than any it is by no means indispensable, Although other,* it possesses certain advantages of symmetry and of generality which The differential notation, is due to Leibniz. are convenient, especially in the study of functions of several varia This notation is founded upon the use of infinitesimals. bles. which approaches zero as small a limit is called an infinitely quantity, or simply an infinitesi the that The condition mal. quantity be variable is essential, for not an infinitesimal unless it is zero.
But the nth increment considered is equal to 0(x + hi, y, z) y, z); and if we apply the law of the mean to this increment, we finally obtain the formula sought. <f>(x, Conversely, the partial derivative fxT^ zr AX hp ki k 2 hi h? as all the increments h, k, It is interesting to the usual definition. function of x and y, no first I approach kg --- A/-/ lr li zero. notice that this definition sometimes more general than + ^(y) is a Then u also has is Suppose, for example, that w =/(x, y) where neither nor ^ has a derivative.
A Course in Mathematical Analysis by Goursat E.