Difference between revisions of "Green's Theorem"

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=== Green's Theorem ===
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[[File:2D loop decomposition.svg|thumb|A demonstration of how a large loop can be decomposed into a family of infinitesimal loops.]]
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Quantifying "circulation density" is best introduced in 2 dimensions. Given a large counter-clockwise oriented loop <math>C</math> that is confined to 2 dimensions, <math>C</math> can be decomposed into a family of infinitesimal loops as shown on the right. Boundaries that are common to adjacent loops cancel each other out due to their opposite orientations, so the total circulation around <math>C</math> is the sum of the circulations around each infinitesimal loop.
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[[File:2D Infinitesimal Loop.svg|thumb|An infinitesimal rectangular loop.]]
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Consider the infinitesimal rectangle <math>R = [x_l,x_u] \times [y_l,y_u]</math>. Let <math>(x_c,y_c) \in R</math> be an arbitrary point inside the rectangle, let <math>\Delta x = x_u - x_l</math> and <math>\Delta y = y_u - y_l</math>, and let <math>\partial R</math> be the counterclockwise boundary of <math>R</math>.
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The circulation around <math>\partial R</math> is approximately (the relative error vanishes as <math>\Delta x, \Delta y \rightarrow 0^+</math>):
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<math> \int_{\mathbf{q} \in \partial R} \mathbf{F}(\mathbf{q}) \cdot d\mathbf{q} \approx
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\mathbf{F}(x_u,y_c) \cdot (+\Delta y \mathbf{j}) +
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\mathbf{F}(x_c,y_u) \cdot (-\Delta x \mathbf{i}) +
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\mathbf{F}(x_l,y_c) \cdot (-\Delta y \mathbf{j}) +
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\mathbf{F}(x_c,y_l) \cdot (+\Delta x \mathbf{i}) </math>
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<math> = F_y(x_u,y_c)\Delta y - F_x(x_c,y_u)\Delta x - F_y(x_l,y_c)\Delta y + F_x(x_c,y_l)\Delta x </math>
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<math> = \left(\frac{F_y(x_u,y_c)-F_y(x_l,y_c)}{\Delta x} - \frac{F_x(x_c,y_u)-F_x(x_c,y_l)}{\Delta y}\right)\Delta x \Delta y</math>
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<math> \approx \left(\frac{\partial F_y}{\partial x}\bigg|_{(x_c,y_c)} - \frac{\partial F_x}{\partial y}\bigg|_{(x_c,y_c)}\right)\Delta x \Delta y</math>
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<math> \approx \iint_R \left(\frac{\partial F_y}{\partial x} - \frac{\partial F_x}{\partial y}\right)dxdy </math>
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As <math>\Delta x, \Delta y \rightarrow 0^+</math>, the relative errors present in the approximations vanish, and therefore, for an infinitesimal rectangle, <math> \int_{\mathbf{q} \in \partial R} \mathbf{F}(\mathbf{q}) \cdot d\mathbf{q} = \iint_{\mathbf{q} \in R} \left(\frac{\partial F_y}{\partial x} - \frac{\partial F_x}{\partial y}\right)dxdy </math>
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<math>\frac{\partial F_y}{\partial x} - \frac{\partial F_x}{\partial y}</math> is the "circulation density" at <math>(x_c, y_c)</math>. Let <math>C</math> be a counter-clockwise oriented loop with interior <math>D</math>. The circulation around loop <math>C</math> is the total circulation contained by <math>D</math>: <math>\int_{\mathbf{q} \in C} \mathbf{F}(\mathbf{q}) \cdot d\mathbf{q} = \iint_{\mathbf{q} \in D} \left(\frac{\partial F_y}{\partial x} - \frac{\partial F_x}{\partial y}\right)dxdy </math>. This is Green's theorem.
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==Resources==
 
* [https://www.youtube.com/watch?v=sDby9VzPoj4 Green's Theorem Part 1] by James Sousa, Math is Power 4U
 
* [https://www.youtube.com/watch?v=sDby9VzPoj4 Green's Theorem Part 1] by James Sousa, Math is Power 4U
 
* [https://www.youtube.com/watch?v=oXISOEqd0X4 Green's Theorem Part 2] by James Sousa, Math is Power 4U
 
* [https://www.youtube.com/watch?v=oXISOEqd0X4 Green's Theorem Part 2] by James Sousa, Math is Power 4U

Revision as of 14:14, 1 October 2021

Green's Theorem

A demonstration of how a large loop can be decomposed into a family of infinitesimal loops.

Quantifying "circulation density" is best introduced in 2 dimensions. Given a large counter-clockwise oriented loop Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C} that is confined to 2 dimensions, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C} can be decomposed into a family of infinitesimal loops as shown on the right. Boundaries that are common to adjacent loops cancel each other out due to their opposite orientations, so the total circulation around is the sum of the circulations around each infinitesimal loop.

An infinitesimal rectangular loop.

Consider the infinitesimal rectangle . Let be an arbitrary point inside the rectangle, let and , and let be the counterclockwise boundary of .

The circulation around is approximately (the relative error vanishes as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta x, \Delta y \rightarrow 0^+} ):

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int_{\mathbf{q} \in \partial R} \mathbf{F}(\mathbf{q}) \cdot d\mathbf{q} \approx \mathbf{F}(x_u,y_c) \cdot (+\Delta y \mathbf{j}) + \mathbf{F}(x_c,y_u) \cdot (-\Delta x \mathbf{i}) + \mathbf{F}(x_l,y_c) \cdot (-\Delta y \mathbf{j}) + \mathbf{F}(x_c,y_l) \cdot (+\Delta x \mathbf{i}) } Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle = F_y(x_u,y_c)\Delta y - F_x(x_c,y_u)\Delta x - F_y(x_l,y_c)\Delta y + F_x(x_c,y_l)\Delta x } Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle = \left(\frac{F_y(x_u,y_c)-F_y(x_l,y_c)}{\Delta x} - \frac{F_x(x_c,y_u)-F_x(x_c,y_l)}{\Delta y}\right)\Delta x \Delta y} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \approx \left(\frac{\partial F_y}{\partial x}\bigg|_{(x_c,y_c)} - \frac{\partial F_x}{\partial y}\bigg|_{(x_c,y_c)}\right)\Delta x \Delta y} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \approx \iint_R \left(\frac{\partial F_y}{\partial x} - \frac{\partial F_x}{\partial y}\right)dxdy }

As Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta x, \Delta y \rightarrow 0^+} , the relative errors present in the approximations vanish, and therefore, for an infinitesimal rectangle, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int_{\mathbf{q} \in \partial R} \mathbf{F}(\mathbf{q}) \cdot d\mathbf{q} = \iint_{\mathbf{q} \in R} \left(\frac{\partial F_y}{\partial x} - \frac{\partial F_x}{\partial y}\right)dxdy }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\partial F_y}{\partial x} - \frac{\partial F_x}{\partial y}} is the "circulation density" at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (x_c, y_c)} . Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C} be a counter-clockwise oriented loop with interior Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle D} . The circulation around loop Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C} is the total circulation contained by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle D} : Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int_{\mathbf{q} \in C} \mathbf{F}(\mathbf{q}) \cdot d\mathbf{q} = \iint_{\mathbf{q} \in D} \left(\frac{\partial F_y}{\partial x} - \frac{\partial F_x}{\partial y}\right)dxdy } . This is Green's theorem.


Resources

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