Green's Theorem - Revision history

Lila at 23:14, 2 November 2021

2021-11-02T23:14:30Z

Lila: /* Relationship to the divergence theorem */

2021-11-02T23:13:27Z

Relationship to the divergence theorem

Lila: /* Relationship to the divergence theorem */

2021-11-02T23:12:39Z

Relationship to the divergence theorem

Lila: /* Proof when D is a simple region */

2021-11-02T23:10:25Z

Proof when D is a simple region

Lila: /* Proof when D is a simple region */

2021-11-02T23:08:36Z

Proof when D is a simple region

Lila: /* Proof when D is a simple region */

2021-11-02T23:07:55Z

Proof when D is a simple region

Lila: /* Proof when D is a simple region */

2021-11-02T23:06:58Z

Proof when D is a simple region

Lila: /* Proof when D is a simple region */

2021-11-02T23:06:21Z

Proof when D is a simple region

Lila: /* Proof when D is a simple region */

2021-11-02T23:05:17Z

Proof when D is a simple region

Lila: /* Proof when D is a simple region */

2021-11-02T23:02:04Z

Proof when D is a simple region

@@ Line 213: / Line 213: @@
 ==Area calculation==
-Green's theorem can be used to compute area by line integral.<ref name="stuart">{{cite book|last=Stewart|first=James|title=Calculus|year=1999|url=https://archive.org/details/calculus00stew|url-access=registration|publisher=Thomson, Brooks/Cole|edition=6th}}</ref> The area of a planar region <math>D</math> is given by
+Green's theorem can be used to compute area by line integral. The area of a planar region <math>D</math> is given by
 :<math>A = \iint_D dA.</math>
@@ Line 221: / Line 221: @@
 :<math>A = \oint_{C} (L\, dx + M\, dy).</math>
-Possible formulas for the area of <math>D</math> include<ref name="stuart" />
+Possible formulas for the area of <math>D</math> include
 :<math>A=\oint_C x\, dy = -\oint_C y\, dx = \tfrac 12 \oint_C (-y\, dx + x\, dy).</math>

@@ Line 198: / Line 198: @@
 ==Relationship to the divergence theorem==
 Considering only two-dimensional vector fields, Green's theorem is equivalent to the two-dimensional version of the divergence theorem:
 <math>\iiint_V\left(\mathbf{\nabla}\cdot\mathbf{F}\right)\,dV= \oiint_{\partial \scriptstyle V} (\mathbf{F}\cdot\mathbf{\hat n})\,dS .</math>

@@ Line 199: / Line 199: @@
 Considering only two-dimensional vector fields, Green's theorem is equivalent to the two-dimensional version of the divergence theorem:
-:{{oiint
+<math>\iiint_V\left(\mathbf{\nabla}\cdot\mathbf{F}\right)\,dV= \oiint_{\partial \scriptstyle V} (\mathbf{F}\cdot\mathbf{\hat n})\,dS .</math>
-| preintegral = <math>\iiint_V\left(\mathbf{\nabla}\cdot\mathbf{F}\right)\,dV=</math>
 where <math>\nabla\cdot\mathbf{F}</math> is the divergence on the two-dimensional vector field <math>\mathbf{F}</math>, and <math>\mathbf{\hat n}</math> is the outward-pointing unit normal vector on the boundary.

@@ Line 31: / Line 31: @@
 where ''g''<sub>1</sub> and ''g''<sub>2</sub> are continuous functions on [''a'', ''b'']. Compute the double integral in (1):
-{{NumBlk|:|<math>
+<math>
 \begin{align}
 \iint_D \frac{\partial L}{\partial y}\, dA
@@ Line 37: / Line 37: @@
 & = \int_a^b \left[L(x,g_2(x)) - L(x,g_1(x)) \right] \, dx.
 \end{align}
-</math>|{{EquationRef|3}}}}
+\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;</math> (3)
-Now compute the line integral in ({{EquationNote|1}}). ''C'' can be rewritten as the union of four curves: ''C''<sub>1</sub>, ''C''<sub>2</sub>, ''C''<sub>3</sub>, ''C''<sub>4</sub>.
+Now compute the line integral in (1). ''C'' can be rewritten as the union of four curves: ''C''<sub>1</sub>, ''C''<sub>2</sub>, ''C''<sub>3</sub>, ''C''<sub>4</sub>.
 With ''C''<sub>1</sub>, use the parametric equations: ''x'' = ''x'', ''y'' = ''g''<sub>1</sub>(''x''), ''a'' ≤ ''x'' ≤ ''b''. Then
@@ Line 55: / Line 55: @@
 Therefore,
-{{NumBlk|:|<math>
+<math>
 \begin{align}
 \int_C L\, dx & = \int_{C_1} L(x,y)\, dx + \int_{C_2} L(x,y)\, dx + \int_{C_3} L(x,y)\, dx + \int_{C_4} L(x,y)\, dx \\
 & = \int_a^b L(x,g_1(x))\, dx - \int_a^b L(x,g_2(x))\, dx .
 \end{align}
-</math>|{{EquationRef|4}}}}
+\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;</math> (4)
-Combining ({{EquationNote|3}}) with ({{EquationNote|4}}), we get ({{EquationNote|1}}) for regions of type I. A similar treatment yields ({{EquationNote|2}}) for regions of type II. Putting the two together, we get the result for regions of type III.
+Combining (3) with (4), we get (1) for regions of type I. A similar treatment yields (2) for regions of type II. Putting the two together, we get the result for regions of type III.
 == Proof for rectifiable Jordan curves ==

@@ Line 21: / Line 21: @@
 and
-<math>\oint_C\ M\, dy = \iint_{D} \left(\frac{\partial M}{\partial x}\right) dA \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;</math> (2)
+<math>\oint_C\ M\, dy = \iint_{D} \left(\frac{\partial M}{\partial x}\right) dA \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;</math> (2)
 are true, then Green's theorem follows immediately for the region D. We can prove the first equation easily for regions of type I, and the second equation for regions of type II. Green's theorem then follows for regions of type III.

@@ Line 17: / Line 17: @@
 If it can be shown that if
-<math>\oint_C L\, dx = \iint_{D} \left(- \frac{\partial L}{\partial y}\right) \;\;\;</math> (1)
+<math>\oint_C L\, dx = \iint_{D} \left(- \frac{\partial L}{\partial y}\right) dA \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;</math> (1)
 and
-<math>\oint_C\ M\, dy = \iint_{D} \left(\frac{\partial M}{\partial x}\right) dA</math> (2)
+<math>\oint_C\ M\, dy = \iint_{D} \left(\frac{\partial M}{\partial x}\right) dA \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;</math> (2)
 are true, then Green's theorem follows immediately for the region D. We can prove the first equation easily for regions of type I, and the second equation for regions of type II. Green's theorem then follows for regions of type III.