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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DF FUNCTIONAL RELATIONS 40 [II, 23 We shall see later the preceding reasoning is no longer applicable.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.

Another special case in which the general formula reduces to a simpler form is that in which u, v, w are integral linear functions In particular, if have, after division , of the independent variables x, y, where the have u= ax v = w= a"x coefficients a, a a x z. + cz+f, + b y + c z +/ + + by -f , c"z +/", b"y , = dv = dw = du a", b, b a dx + a dx -f- a"dx + are constants. , b b + dy + + dy b"dy c dz, c dz, c"dz, the differentials of higher order d u, dn v, dn iv, where n>l, vanish.

Equations (8) ; I, THE DIFFERENTIAL NOTATION 16] In general, if 25 the nth total differential be obtained in any whatever, = 2 Cpqr w d" dx" dy" dz way r ; Cyqr then the coefficients are respectively equal to the corresponding nth derivatives multiplied by certain numerical factors. Thus all these derivatives are determined at once. We shall have occasion to use these facts presently. written down as follows w : dia_dFdii dFdv dFdw du dx dv dx dw dx _d_F_d_u du dy d_F_d_v_ ___ dy dv dy dw dy dz du dz do dz dw dw dF du dF dv dF dw du dv dt dw dx d> = Let w F(u, v, w~) being themselves functions of the The partial derivatives may then be 16.

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A Course in Mathematical Analysis by Goursat E.

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