Green's Theorem

Green's Theorem

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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 $C$ that is confined to 2 dimensions, $C$ 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 $C$ is the sum of the circulations around each infinitesimal loop.

An infinitesimal rectangular loop.

Consider the infinitesimal rectangle $R=[x_{l},x_{u}]\times [y_{l},y_{u}]$. Let $(x_{c},y_{c})\in R$ be an arbitrary point inside the rectangle, 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 \Delta x = x_u - x_l} and $\Delta y=y_{u}-y_{l}$, and 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 \partial R} be the counterclockwise boundary of $R$.

The circulation around 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 \partial R} 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 $C$ 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

• Green's Theorem Part 1 by James Sousa, Math is Power 4U
• Green's Theorem Part 2 by James Sousa, Math is Power 4U
• Evaluate a Line Integral Using Green's Theorem by James Sousa, Math is Power 4U
• Use Green's Theorem to Evaluate a Line Integral on a Rectangle by James Sousa, Math is Power 4U
• Use Green's Theorem to Evaluate a Line Integral of a Vector Field on a Circle by James Sousa, Math is Power 4U
• Use Green's Theorem to Evaluate a Line Integral Using Polar Coordinates by James Sousa, Math is Power 4U
• Use Green's Theorem to Evaluate a Line Integral with Negative Orientation by James Sousa, Math is Power 4U
• Use Green's Theorem to Determine the Area of a Region Enclosed by a Curve by James Sousa, Math is Power 4U
• Determining Area Using Line Integrals by James Sousa, Math is Power 4U
• Flux Form of Green's Theorem by James Sousa, Math is Power 4U
• Determine the Flux of a 2D Vector Field Across a Rectangle Using Green's Theorem by James Sousa, Math is Power 4U
• Determine the Flux of a 2D Vector Field Across a Parabolic Region Using Green's Theorem by James Sousa, Math is Power 4U
• Determine the Flux of a 2D Vector Field Using Green's Theorem (Hole) by James Sousa, Math is Power 4U

• Green's Theorem by Patrick JMT

• Green's Theorem (One Region) by Krista King
• Green's Theorem (Two Regions) by Krista King

• Green's Theorem Example 1 by Khan Academy
• Green's Theorem Example 2 by Khan Academy