A Course of Mathematical Analysis, Part II by A. F. Bermant
By A. F. Bermant
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Additional info for A Course of Mathematical Analysis, Part II
Example text
E. x (Fig. 14a). It is clear from the figure that in the present case I~ (xo' Yo) < o. e. f~ (xo, Yo) = tan {3 (Fig. 14b). It is clear from the figure that in the present case f~(xo, Yo) > o. I:, 145. Differentials 1. PARTIAL DIFFERENTIALS. The increment that the function z = I(x, y) receives when only one of the variables alters is termed the partial increment of the function with respect to that variable. The following notations are used: Llzz = I(x + Llx, Llvz == I (x, y y) - I(x, y), + LI y) - I (x, y).
For we have, since the function is continuous, lim f(x, y, z, ... , t) P ..... p. e. limf(x, y, z, , .. , t) = f(x o, Yo' zo, ... , limz, ... , limt), which can be written as lim/(P) =/(limP). Thus the symbol for the limit and the symbol for a continuous function can be interchanged. FUNOTIONS OF SEVERAL VARIABLES 21 143. 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.
E. as Llx -+ 0, Lly -? 0). Hence (see Sec. hand side of equation (*) is the differential dz: dz = (f~