Gradient Fields, Conservative Vector Field, Fundamental Theorem for Line Integrals

9 minute read

๋ณต์ˆ˜์ „๊ณตํ•˜๊ณ  ์žˆ๋Š” ์ˆ˜ํ•™๊ณผ์˜ ์กธ์—…์‹œํ—˜์„ ์œ„ํ•ด ํ•™๋ถ€ ์ˆ˜ํ•™ ๊ณผ๋ชฉ๋“ค์„ ๋‹ค์‹œ ๊ณต๋ถ€ํ•˜๊ณ  ์žˆ์Šต๋‹ˆ๋‹ค. ๋ฏธ์ ๋ถ„ํ•™ ํฌ์ŠคํŠธ ์ „์ฒด ๋ณด๊ธฐ

Gradient FieldsPermalink

์–ด๋–ค 2๋ณ€์ˆ˜ ํ•จ์ˆ˜ f(x,y)๊ฐ€ ์žˆ๋‹ค๊ณ  ํ•˜์ž. ์ด์ „ ์ฑ•ํ„ฐ์—์„œ ์ด 2๋ณ€์ˆ˜ ํ•จ์ˆ˜์˜ ํ•œ ์  (x0,y0)์—์„œ ์ •์˜ํ•œ Gradient Vector๋ฅผ ๊ธฐ์–ตํ•˜๋Š”๊ฐ€?

โˆ‡f(x,y)=fx(x,y)i+fy(x,y)j

์ด๊ฒƒ์„ ํ•œ ์ ์ด ์•„๋‹ˆ๋ผ ํ•จ์ˆ˜ f(x,y)์˜ ์ •์˜์—ญ ์ „์ฒด์—์„œ ์ •์˜ํ•œ ๊ฒƒ์ด โ€œGradient Fieldโ€๋‹ค. ์ด๊ฒƒ์€ ์Šค์นผ๋ผ ํ•จ์ˆ˜์ธ z=f(x,y)๋กœ๋ถ€ํ„ฐ ์œ ๋„๋˜๋Š” ๋ฒกํ„ฐ ํ•„๋“œ(= ๋ฒกํ„ฐ ํ•จ์ˆ˜)๋‹ค.

Conservative Vector Field, and Potential FunctionPermalink

๋ฐ˜๋Œ€๋กœ ์–ด๋–ค ๋ฒกํ„ฐ ํ•„๋“œ๋Š” ๊ทธ ์›๋ณธ์ด ์–ด๋–ค ์Šค์นผ๋ผ ํ•จ์ˆ˜์ธ ๊ฒƒ๋“ค์ด ์žˆ๋‹ค. ์ด๋Ÿฐ ๋ฒกํ„ฐ ํ•„๋“œ๋ฅผ โ€œConservative Vector Fieldโ€๋ผ๊ณ  ํ•œ๋‹ค. ์ด๋“ค์€ Gradient โˆ‡ ๋˜๊ธฐ ์ „์˜ ์›์‹œ ์Šค์นผ๋ผ ํ•จ์ˆ˜๋ฅผ ์ฐพ์„ ์ˆ˜๋„ ์žˆ๋‹ค.

A vector field F is called a โ€œconservative vector fieldโ€ if it is the gradient of some scalar function.

์ด๋Ÿฐ Conservative Vector Field์˜ ์›์‹œ ์Šค์นผ๋ผ ํ•จ์ˆ˜๋Š” โ€œPotential Functionโ€œ๋ผ๊ณ  ๋ถ€๋ฅธ๋‹ค.

Line IntegralsPermalink

์–ด๋–ค ๊ณก์„ ์„ ๋”ฐ๋ผ ํ•จ์ˆ˜ f(x,y)์— ์ ๋ถ„์„ ์ˆ˜ํ–‰ํ•˜๋Š” ๊ฒƒ์„ โ€œ์„ ์ ๋ถ„โ€์ด๋ผ๊ณ  ํ•œ๋‹ค.

Scalar Line Integral

โˆซCf(x,y)ds=โˆซCf(x(t),y(t))โ‹…xโ€ฒ(t)2+yโ€ฒ(t)2dt

Vector Line Integral

โˆซCF(x,y)โ‹…dr=โˆซCF1dx+F2dy+F3dz

์„ ์ ๋ถ„ ๋ถ€๋ถ„์€ ๊ณต๋ถ€ํ•˜๋‹ค๊ฐ€ ๋„ˆ๋ฌด ํ—ท๊ฐˆ๋ ค์„œ ๋ณ„๋„ ํฌ์ŠคํŠธ๋กœ ์ •๋ฆฌํ–ˆ๋‹ค. ์„ ์ ๋ถ„์— ๋Œ€ํ•œ ์ž์„ธํ•œ ๋‚ด์šฉ์€ ์•„๋ž˜ ํฌ์ŠคํŠธ ์ฐธ๊ณ  ใ…Žใ…Ž

โžก๏ธ Arc Length์™€ Line Integral

Fundamental Theorem for Line IntegralsPermalink

Let C be a smooth curve given by the vector function r(t) for aโ‰คtโ‰คb.

Let f(x,y) be a differentiable function of two or three variables whose gradient vector โˆ‡f is continuous on C.

Then,

โˆซCโˆ‡fโ‹…dr=f(r(b))โˆ’f(r(a))

f(x)์˜ ์ ๋ถ„์„ ์‹œ์ž‘์ ๊ณผ ๋์ ์—์„œ์˜ ์›์‹œํ•จ์ˆ˜ F(x)์˜ ๊ฐ’์˜ ์ฐจ์ด๋กœ ๊ตฌํ•  ์ˆ˜ ์žˆ๋‹ค๋Š” ๋ฏธ์ ๋ถ„ํ•™์˜ ๊ธฐ๋ณธ์ •๋ฆฌ์˜ ์„ ์ ๋ถ„ ๋ฒ„์ „์ด๋‹ค. ์„ ์ ๋ถ„์—์„œ๋Š” ์ ๋ถ„ํ•˜๋ ค๋Š” ํ•จ์ˆ˜๊ฐ€ Gradient Field๋ผ๋ฉด, ์‹œ์ž‘์ ๊ณผ ๋์ ์—์„œ ์›์‹œํ•จ์ˆ˜์˜ ๊ฐ’์˜ ์ฐจ์ด๋กœ ์ ๋ถ„์„ ๊ตฌํ•  ์ˆ˜ ์žˆ๋‹ค๋Š” ์ •๋ฆฌ์ด๋‹ค. ์ ๋ถ„์ด ๋ฌด์ง€๋ฌด์ง€ ์‰ฌ์›Œ์ง„๋‹ค๋Š” ๋ง!!

์ ๋ถ„ํ•˜๋ ค๋Š” ํ•จ์ˆ˜๊ฐ€ Gradient Field๋ผ๋Š” ๋ง์€ ๊ณง, ๊ทธ ํ•จ์ˆ˜๊ฐ€ โ€œConservative Vector Fieldโ€์ž„์„ ๋งํ•œ๋‹ค. ์ฆ‰, Conservative Vector Field์—์„œ ์„ฑ๋ฆฝํ•˜๋Š” ์ •๋ฆฌ๊ฐ€ โ€œ์„ ์ ๋ถ„์˜ ๊ธฐ๋ณธ์ •๋ฆฌโ€์ธ ๊ฒƒ. ์ ๋ถ„๊ฐ’์„ ๊ณ„์‚ฐํ•˜๊ธฐ ์œ„ํ•ด์„œ ์ ์ ˆํ•œ Potential Function๋งŒ ์ฐพ์œผ๋ฉด ๋œ๋‹ค.

Work done by Gravitational FieldPermalink

Find the work done by the gravitational field

F(x)=โˆ’mMGโ€–xโ€–3x

in moving a particle with mass m from (0,0,0) to (1,1,1) along a piecewise-smooth curve C.

์œ„์˜ ์„ ์ ๋ถ„์„ ๊ณ„์‚ฐํ•˜๊ธฐ ์œ„ํ•ด์„œ ๋ฌธ์ œ์— ์ œ์‹œ๋œ piecewise-smooth curve C๋ฅผ ์ฐพ์„ ํ•„์š˜ ์—†๋‹ค. F(x)์˜ Potential Function๋งŒ ์ฐพ์„ ์ˆ˜ ์žˆ๋‹ค๋ฉด, ์„ ์ ๋ถ„์˜ ๊ธฐ๋ณธ์ •๋ฆฌ๋กœ ์‹œ์ ๊ณผ ์ข…์ ์—์„œ์˜ ๊ฐ’์œผ๋กœ ์ ๋ถ„์„ ๊ณ„์‚ฐํ•˜๋ฉด ๋˜๊ธฐ ๋•Œ๋ฌธ.

์ค‘๋ ฅ์žฅ F(x)์˜ potential function์€ ์•„๋ž˜์™€ ๊ฐ™์ด ์ •์˜ํ•  ์ˆ˜ ์žˆ๋‹ค.

f(x,y,z)=mMGx2+y2+z2

potential function์„ ์ฐพ์•˜์œผ๋‹ˆ ์„ ์ ๋ถ„์˜ ๊ธฐ๋ณธ์ •๋ฆฌ๋กœ ์ ๋ถ„์„ ๊ณ„์‚ฐํ•ด๋ณด์ž.

W=โˆซCFโ‹…dr=โˆซCโˆ‡fโ‹…dr=f(1,1,1)โˆ’f(0,0,0)=mMG3

Independent of PathPermalink

์‹œ์ ๊ณผ ์ข…์ ์€ ๊ฐ™์ง€๋งŒ, ๊ฒฝ๋กœ๊ฐ€ ์„œ๋กœ ๋‹ค๋ฅธ piecewise-smooth curve C1, C2๊ฐ€ ์žˆ๋‹ค๊ณ  ํ•˜์ž.

์ผ๋ฐ˜์ ์ธ ๋ฒกํ„ฐ ํ•„๋“œ์—์„œ๋Š” ๋‘ ์„ ์ ๋ถ„์˜ ๊ฐ’์ด ๊ฐ™์ง€ ์•Š๋‹ค.

โˆซC1Fโ‹…drโ‰ โˆซC2Fโ‹…dr

๊ทธ๋Ÿฌ๋‚˜ ๋ฒกํ„ฐ ํ•„๋“œ๊ฐ€ Conservative Vector Field๋ผ๋ฉด, ๋‘ ์„ ์ ๋ถ„์˜ ๊ฐ’์ด ๊ฐ™์•„์ง„๋‹ค.

โˆซC1โˆ‡fโ‹…dr=โˆซC2โˆ‡fโ‹…dr

์™œ๋ƒํ•˜๋ฉด, ๋‘ ์„ ์ ๋ถ„์ด ์‹œ์ , ์ข…์ ์—์„œ์˜ potential function์˜ ๊ฐ’ ์ฐจ์ด๋กœ ๊ณ„์‚ฐ๋˜๊ธฐ ๋•Œ๋ฌธ์ด๋‹ค.

์ด๊ฒƒ์„ ์ •๋ฆฌํ•œ ๊ฒƒ์ด ์•„๋ž˜์˜ ๋ฌธ์žฅ์ด๋‹ค.

Line integrals of a continuous conservative vector field with a differentiable potential function are โ€œindependent of pathโ€.

On a Closed CurvePermalink

์ด๋ฒˆ์—๋Š” ๋‹ซํžŒ ๊ณก์„  C์—์„œ Conservative Vector Field์˜ ์ ๋ถ„์„ ์‚ดํŽด๋ณด์ž. ๊ฒฐ๋ก ๋ถ€ํ„ฐ ๋งํ•˜๋ฉด, ํ๊ณก์„ ์—์„œ์˜ ์„ ์ ๋ถ„์˜ ๊ฐ’์€ ํ•ญ์ƒ 0์ด๋‹ค.

โˆซCFโ‹…dr=0

์œ„์˜ ์„ ์ ๋ถ„์„ ๊ฒฝ๋กœ C1, C2๋กœ ๋ถ„ํ• ํ•˜์—ฌ ์ƒ๊ฐํ•˜๋ฉด ์•„๋ž˜์™€ ๊ฐ™๊ธฐ ๋•Œ๋ฌธ.

โˆซCFโ‹…dr=โˆซC1Fโ‹…dr+โˆซC2Fโ‹…dr=โˆซC1Fโ‹…drโˆ’โˆซโˆ’C2Fโ‹…dr=0

(์š”๊ธฐ์—์„œ opposite direction ์ ๋ถ„ ๋ณผ ๋•Œ, ์Šค์นผ๋ผ ์„ ์ ๋ถ„์ด๋ž‘ ํ—ท๊ฐˆ๋ ค์„œ ํ•œ์ฐธ ๊ณ ๋ฏผํ•จโ€ฆ ใ…‹ใ…‹)

TheoremsPermalink

Independent of Path implies ConservativePermalink

๋งŒ์•ฝ ์ฃผ์–ด์ง„ ๋ฒกํ„ฐ ํ•„๋“œ์˜ ์ ๋ถ„์ด ์ฃผ์–ด์ง„ ๋„๋ฉ”์ธ D์—์„œ ๋ชจ๋‘ independent of path๋ผ๋ฉด, ํ•ด๋‹น ๋ฒกํ„ฐ ํ•„๋“œ๋Š” ๋„๋ฉ”์ธ D ์œ„์—์„œ Conservative Field์ด๋‹ค.

๋ณธ๋ž˜ Conservative Field๋ฉด, Independent of path๋ฅผ ๋งŒ์กฑํ•˜๋Š”๋ฐ, ๊ทธ ์—ญ ๋ช…์ œ๋„ ์„ฑ๋ฆฝํ•จ์„ ๋งํ•œ๋‹ค.

If Conservative Field, thenPermalink

๋งŒ์•ฝ ๋ฒกํ„ฐ ํ•„๋“œ F=P(x,y)i+Q(x,y)j๊ฐ€ conservative vector field์ด๊ณ , P, Q ํ•จ์ˆ˜๊ฐ€ ๋„๋ฉ”์ธ D ์œ„์—์„œ continuous first-order partial derivative๋ฅผ ๊ฐ€์ง„๋‹ค๋ฉด, ์•„๋ž˜ ๋“ฑ์‹์„ ๋งŒ์กฑํ•œ๋‹ค.

โˆ‚Pโˆ‚y=โˆ‚Qโˆ‚x


์ฆ๋ช…์€ ๊ฐ„๋‹จํ•œ๋ฐ, ๋ฒกํ„ฐ ํ•„๋“œ F๊ฐ€ conservative ํ•˜๋ฏ€๋กœ, ์•„๋ž˜ ์‹์„ ๋งŒ์กฑํ•˜๋Š” potential function f๊ฐ€ ์กด์žฌํ•œ๋‹ค.

F=โˆ‡f

๋”ฐ๋ผ์„œ ๊ฐ ์„ฑ๋ถ„ P, Q๋Š” ์•„๋ž˜์™€ ๊ฐ™์ด 1์ฐจ ํŽธ๋ฏธ๋ถ„์œผ๋กœ ์ •์˜๋œ๋‹ค.

P(x,y)=โˆ‚fโˆ‚xQ(x,y)=โˆ‚fโˆ‚y

์ด์ œ, P, Q ์„ฑ๋ถ„์— ๋‹ค์‹œ y์™€ x์— ๋Œ€ํ•ด ํŽธ๋ฏธ๋ถ„ ํ•˜๋ฉด ์•„๋ž˜์™€ ๊ฐ™๋‹ค.

โˆ‚Pโˆ‚y=โˆ‚fโˆ‚yโˆ‚x=โˆ‚fโˆ‚xโˆ‚y=โˆ‚Qโˆ‚x

โ—ผ

Condition of conservative fieldPermalink

๋ฐ”๋กœ ์œ„์—์„œ ์‚ดํŽด๋ณธ ๋ช…์ œ์˜ ์—ญ ๋ช…์ œ๊ฐ€ ์–ธ์ œ ์„ฑ๋ฆฝํ•˜๋Š”์ง€๋„ ์‚ดํŽด๋ณด์ž.

Let F=Pi+Qj be a vector field on an open simply-connected region D.

Suppose that P and Q have continuous first-order derivatives and satisfy โˆ‚Pโˆ‚y=โˆ‚Qโˆ‚x throughout D.

Then, F is conservative.

์ฃผ์–ด์ง„ ๋ฒกํ„ฐ ํ•„๋“œ๊ฐ€ Conservative ํ•œ์ง€ ํŒ๋‹จํ•˜๋Š” ๋˜ ๋‹ค๋ฅธ ๋ฐฉ๋ฒ•์ด๋‹ค. Conservative ์—ฌ๋ถ€๋ฅผ ํŒ๋‹จํ•˜๊ธฐ ์œ„ํ•ด์„  P, Q ์„ฑ๋ถ„์˜ ํŽธ๋ฏธ๋ถ„ ๊ฐ’์ด ์ผ์น˜ํ•œ์ง€๋ฅผ ํ™•์ธํ•˜๋ผ๋Š” ๋ง.


๊ทธ๋Ÿฐ๋ฐ ์—ฌ๊ธฐ์„œ ์ฒ˜์Œ ๋“ฑ์žฅํ•œ ๊ฐœ๋…์ด โ€œsimply-connected regionโ€์ด๋‹ค. ๋Œ€์ถฉ ๋‚˜๋ˆ ์„œ ์„ค๋ช…ํ•˜๋ฉด,

Gilbert Strang - Calculus Vol 3.

โ€œsimple curveโ€๋Š” ๊ณก์„  ์ž์ฒด๊ฐ€ ์ž๊ธฐ ์ž์‹ ๊ณผ ๋‹ค์‹œ ๋งŒ๋‚˜์ง€ ์•Š๋Š” ๋‚˜์ด์Šคํ•œ ๊ณก์„ ์„ ๋งํ•œ๋‹ค.


โ€œsimply-connected regionโ€์€ ์˜์—ญ ์œ„์— ๊ทธ๋ฆด ์ˆ˜ ์žˆ๋Š” ๋ชจ๋“  simple closed curve ์•ˆ์˜ ์ ์ด ๋ชจ๋‘ ์˜์—ญ D์— ์†ํ•˜๋Š” ์ ๋“ค์ธ ๊ฒฝ์šฐ๋ฅผ ๋งํ•œ๋‹ค.

Gilbert Strang - Calculus Vol 3.

๋งŒ์•ฝ ์˜์—ญ ์•ˆ์— ๊ตฌ๋ฉ(hole)์ด ์žˆ๋‹ค๋ฉด, ๊ทธ ๊ตฌ๋ฉ์„ ๋‘˜๋Ÿฌ์‹ธ๋Š” simple closed curve๊ฐ€ ๋งŒ๋“œ๋Š” ์˜์—ญ ์•ˆ์—๋Š” ์˜์—ญ D์— ์†ํ•˜๋Š” ์ ๋„ ์žˆ๊ฒ ์ง€๋งŒ, ์†ํ•˜์ง€ ์•Š๋Š” ์ ๋„ ์ƒ๊ธด๋‹ค. ๋”ฐ๋ผ์„œ 2์ฐจ์›์—์„œ๋Š” ๊ตฌ๋ฉ ์—†๋Š” ์˜์—ญ์„ ์ผ์ปซ๋Š”๋‹ค๊ณ  ๋ณผ ์ˆ˜ ์žˆ๋‹ค.

Categories:

Updated: