Advanced Calculus a Text Upon Select Parts of Differential Calculus Differenti
Advanced Calculus a Text Upon Select Parts of Differential Calculus Differenti
Edwin Bidwell Wilson
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125. The conditions that P' v and Q' T be continuous and equal, which insures independence of the path for the line integral of Pdx + Qdy, need to be examined more closely. Consider two examples : First | Pdx + Qdy = I ' - dx -\ dy, J J x- + y- x- + y- where + //" x- + y- cP y 2 — x- dQ y' 2 — x- cy (x 2 + y 2 ) 2 ' ex (x 2 + y 2 ) 2 It appears formally that P' — Q' r . If the integral be calculated around a square of side 2 a surrounding the origin, the result is X + « + adx r + " ady r - a — ...adx r - a — ady _ r + " adx -a x 2 + a 2 J -a a 2 + y 2 J+u x 2 + a 2 «/ + « a 2 + y 2 J- a x 2 + a 2 „ C + a ady . R + " adl- . Ir „ . + 2 I — = 4 j — = 4- = 27T5^0. J -a a 2 + y 2 J -a £- + a- 2 302 INTEGRAL CALCULUS The integral fails to vanish around the closed path. The reason is not far to seek, the derivatives P' y and Q' x are not defined for (0, 0), and cannot be so defined as to be continuous functions of (x, y) near the origin. As a matter of fact r
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