1. In the plane, let D be a simply-connected region whose boundary ðD is a piecewise C', simple, closed curve,oriented counterclockwise. Let î be the outward unit normal vector to D. Given two functions, f(x,y) andg(x, y), both C2 on an open set containing the domain D, show that:(a)ds = -CCfor any piece-wise C1 closed curve C in the domain D.(b)II. v*9) dzdy = f (sv9) · îû ds -where V'g = V · Vg = gxx + gy ·(c)/I. (SV°g -(SV°9 – gv²f) dædy = f (v9 - gvs)în ds

Answered: Image /qna-images/answer/99ca48dd-dfda-4727-8b06-a1389a8299e0.jpg

1. In the plane, let D be a simply-connected region whose boundary OD is a piecewise C, simple, closed curve, oriented counterclockwise. Let în be the outward unit normal vector to D. Given two functions, f(x,y) and g(x, y), both C² on an open set containing the domain D, show that: (a) · ds = - · dš, C C for any piece-wise C' closed curve C in the domain D. (b) II, uv°9) dædy = f (s79 i ·în ds – ||. (Fs . ÿ9) dædy where V²g = V · Vg = gxx + Iyy· (c) 72 f) dadau - (E,

1. In the plane, let D be a simply-connected region whose boundary OD is a piecewise C, simple, closed curve, oriented counterclockwise. Let în be the outward unit normal vector to D. Given two functions, f(x,y) and g(x, y), both C² on an open set containing the domain D, show that: (a) · ds = - · dš, C C for any piece-wise C' closed curve C in the domain D. (b) II, uv°9) dædy = f (s79 i ·în ds – ||. (Fs . ÿ9) dædy where V²g = V · Vg = gxx + Iyy· (c) 72 f) dadau - (E,

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter3: Functions And Graphs

Section3.3: Lines

Problem 30E

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