By Goursat E.
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Additional info for A Course in Mathematical Analysis
11*. )" x+! ] 12. Every function of the form rx The function z + ^. = x0(x + y) <f> = z and whatever be the functions 13. 2 and \f/. + ty* = + y^(x + y) -2s+t= r whatever be the functions x$(y/x) + $ (y/x) 2 sxy satisfies the 0, satisfies 0, the equation equation DERIVATIVES AND DIFFERENTIALS 34 14. The function =f[x + z satisfies the </>(y)] equation ps [I, = qr, Exs whatever be the functions / and 0. 15. The function z = + rx 2 and <j> Show y where fa + 2 sxy whatever be the functions 16.
Together with its first partial derivatives, is continuous at a point Let z be the corresponding value of z, (^o? yo) of the xy plane. and AT (cr 7/0 the corresponding point on the surface S. If ) , > the equations *=/(*), (3) z/ = = * <KO> ^(9 M the represent a curve C on the surface S through the point three functions f(f), which we shall suppose continuous <j>(t), and differentiable, must reduce to x y t. The tangent , , of the parameter value t M given by the equations is x Since the curve must hold z C lies on the surface t; z , respectively, for some to this curve at the point 5) ( Y x for all values of , "A(0> that S, is, * the equation \j/(t)=F[f(t~), this relation .
Exists one , = = =z x y The derivative Fz not being zero for x y z Since is that it F, Fx positive. suppose, for defmiteness, I sets of values x, y, z (2) and , , , let Fv F , z us are us choose a positive are continuous for all functions four so small that these supposed continuous in the neighborhood, number , \x-x which satisfy the relations \y-y \<l, \*-z \<l, that, for these sets of values of x, y, F z *Iu a recent let (x,y,z} > <l, z, P, Mathematique de France, Vol.
A Course in Mathematical Analysis by Goursat E.