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By A. F. Bermant

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Extra resources for A Course of Mathematical Analysis, Part II

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We describe z as an explicit function of two independent variables x and y if it is given by an equation between x, y and z which is solved for z. An insoluble equation between three variables can, however, define one of them asa function of the other two. In fact, the general form of equation between x, y and z is f(x, y, z) = o. On substituting any given values x = xo' y = Yo for x and y, we obtain an equation in z: f(x o, Yo' z) = 0, from which the value (or values) of z can be found corresponding to x· Xo and y = yo' Hence z is given as a definite function of x and y by the equation f(x, y, z) = O.

Straight line MoT,,). e. LI z, is represented by the segment M1Tl lying between the surface S and the tangent plane T. LI z measures the distance from the surface to the tangent plane with respect to the z axis. It will be seen that this distance is an infinitesimal of higher order than the distance e = POP1 · 147. Application of the Differential to Approximations. e. we neglect the term ()(. in the right-hand side of the strict equation 37 FUNCTIONS OF SEVERAL VARIABLES we obtain the approximate equation I(x, y) - I(x o' Yo) ~ f~(xo, Yo) (x - xo) + f;(x o, Yo) (y - Yo), or f(x, y) ~ f(x o' Yo) + I~(xo, Yo) (x - xo) + I; (xo' Yo) (y - Yo), (*) expressing the given function as a linear function of the independent variables.

The Behaviour of a Function. Level Lines. The study of a function of two independent variables can be reduced by various means to the study of a function of a single independent variable. I. g. e. in some neighbourhood of this point. We draw a straight line in any direction through the point Po(xo , Yo) in the Oxy plane; its equation will be x - Xo = e cos IX, Y - Yo = e sin IX, (*) where e(e > 0) is the distance of the point P (x,y) from Po (xo' Yo)' whilst IX is the angle between the straight line and the positive z direction of 0 x.

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