By Goursat E.

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DF FUNCTIONAL RELATIONS 40 [II, 23 We shall see later the preceding reasoning is no longer applicable. to the various curves which lie on that the tangents (Chapter III) the surface and which pass through not a plane. 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.

Let w =f(x, 15. Total differentials. three independent variables x, o du * This disadvantage = o / dx ^ex + y, z) be a function of the The expression y, z. Q / / ^ dy + dz -^ dz dy - would seem completely to offset the advantage mentioned TRANS. uy, etc. t This use of the letter d to denote the partial derivatives of a function of several variables is due to Jacob! Before his time the same letter d was used as is used for above. the derivatives of a function of a single variable. I, THE DIFFERENTIAL NOTATION 15] 23 where dx, dy, dz are three fixed called the total differential of otherwise which are arbitrary, assigned to the three increments, is o>, independent variables The three products x, y, z.

2) a n may be , zeros. 19. Applications. Let y f(x) be the equation of a plane curve C with The equation of the tangent at a point respect to a set of rectangular axes. M(x, is y) Y-y = y The slope of the normal, tangency, l/y is ; which (X-x). perpendicular to the tangent at the point of and the equation of the normal is, therefore, is N P Let be the foot of the ordinate of the point Jlf, and let T and be the points of intersection of the x axis with the tangent and the normal, respectively. The distance is called the subnormal PN ; the subtangent; MN, the normal; and T, the tangent.