Difference between revisions of "Conservative Vector Fields"

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=== Conservative vector fields ===
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A vector field <math>\mathbf{F}</math> for which <math>\nabla \times \mathbf{F} = \mathbf{0}</math> at all points is an "<b>conservative</b>" vector field. <math>\mathbf{F}</math> can also be referred to as being "<b>irrotational</b>" since the gain around any closed curve is always 0. 
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A key property of a conservative vector field <math>\mathbf{F}</math> is that the gain of <math>\mathbf{F}</math> along a continuous curve is purely a function of the curve's end points. If <math>C_1</math> and <math>C_2</math> are two continuous curves which share the same starting point <math>\mathbf{q}_0</math> and end point <math>\mathbf{q}_1</math>, then <math>\int_{\mathbf{q} \in C_1} \mathbf{F}(\mathbf{q}) \cdot d\mathbf{q} = \int_{\mathbf{q} \in C_2} \mathbf{F}(\mathbf{q}) \cdot d\mathbf{q}</math>. In other words, the gain is purely a function of <math>\mathbf{q}_0</math> and <math>\mathbf{q}_1</math>. This property can be derived from Stokes' theorem as follows:
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Invert the orientation of <math>C_2</math> to get <math>-C_2</math> and combine <math>C_1</math> and <math>-C_2</math> to get a continuous closed curve <math>C_3 = C_1 - C_2</math>, linking the curves together at the endpoints <math>\mathbf{q}_0</math> and <math>\mathbf{q}_1</math>. Let <math>\sigma</math> denote a surface for which <math>C_3</math> is the counterclockwise oriented boundary.
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Stokes' theorem states that <math>\int_{\mathbf{q} \in C_3} \mathbf{F}(\mathbf{q}) \cdot d\mathbf{q} = \iint_{\mathbf{q} \in \sigma} (\nabla \times \mathbf{F})|_\mathbf{q} \cdot \mathbf{dS} = 0</math>. The gain around <math>C_3</math> is the gain along <math>C_1</math> minus the gain along <math>C_2</math>: <math>\int_{\mathbf{q} \in C_3} \mathbf{F}(\mathbf{q}) \cdot d\mathbf{q} = \int_{\mathbf{q} \in C_1} \mathbf{F}(\mathbf{q}) \cdot d\mathbf{q} + \int_{\mathbf{q} \in -C_2} \mathbf{F}(\mathbf{q}) \cdot d\mathbf{q}</math> <math> = \int_{\mathbf{q} \in C_1} \mathbf{F}(\mathbf{q}) \cdot d\mathbf{q} - \int_{\mathbf{q} \in C_2} \mathbf{F}(\mathbf{q}) \cdot d\mathbf{q}</math>. Therefore:
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<math>\int_{\mathbf{q} \in C_3} \mathbf{F}(\mathbf{q}) \cdot d\mathbf{q} = 0</math> <math>\implies \int_{\mathbf{q} \in C_1} \mathbf{F}(\mathbf{q}) \cdot d\mathbf{q} = \int_{\mathbf{q} \in C_2} \mathbf{F}(\mathbf{q}) \cdot d\mathbf{q}</math>
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==Resources==
 
<strong>Conservative Vector Fields</strong>
 
<strong>Conservative Vector Fields</strong>
  

Revision as of 20:28, 10 October 2021

Conservative vector fields

A vector field for which at all points is an "conservative" vector field. can also be referred to as being "irrotational" since the gain around any closed curve is always 0.

A key property of a conservative vector field 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 \mathbf{F}} is that the gain of 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 \mathbf{F}} along a continuous curve is purely a function of the curve's end points. If 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_1} and 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_2} are two continuous curves which share the same starting point 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 \mathbf{q}_0} and end point 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 \mathbf{q}_1} , then 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_1} \mathbf{F}(\mathbf{q}) \cdot d\mathbf{q} = \int_{\mathbf{q} \in C_2} \mathbf{F}(\mathbf{q}) \cdot d\mathbf{q}} . In other words, the gain is purely a function of 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 \mathbf{q}_0} and 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 \mathbf{q}_1} . This property can be derived from Stokes' theorem as follows:

Invert the orientation of 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_2} to get 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_2} and combine 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_1} and 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_2} to get a continuous closed curve 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_3 = C_1 - C_2} , linking the curves together at the endpoints 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 \mathbf{q}_0} and 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 \mathbf{q}_1} . 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 \sigma} denote a surface for which 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_3} is the counterclockwise oriented boundary.

Stokes' theorem states that 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_3} \mathbf{F}(\mathbf{q}) \cdot d\mathbf{q} = \iint_{\mathbf{q} \in \sigma} (\nabla \times \mathbf{F})|_\mathbf{q} \cdot \mathbf{dS} = 0} . The gain 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 C_3} is the gain along 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_1} minus the gain along 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_2} : 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_3} \mathbf{F}(\mathbf{q}) \cdot d\mathbf{q} = \int_{\mathbf{q} \in C_1} \mathbf{F}(\mathbf{q}) \cdot d\mathbf{q} + \int_{\mathbf{q} \in -C_2} \mathbf{F}(\mathbf{q}) \cdot d\mathbf{q}} 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_1} \mathbf{F}(\mathbf{q}) \cdot d\mathbf{q} - \int_{\mathbf{q} \in C_2} \mathbf{F}(\mathbf{q}) \cdot d\mathbf{q}} . Therefore:

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_3} \mathbf{F}(\mathbf{q}) \cdot d\mathbf{q} = 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 \implies \int_{\mathbf{q} \in C_1} \mathbf{F}(\mathbf{q}) \cdot d\mathbf{q} = \int_{\mathbf{q} \in C_2} \mathbf{F}(\mathbf{q}) \cdot d\mathbf{q}}


Resources

Conservative Vector Fields


Finding a Potential Function of a Conservative Vector Field


The Fundamental Theorem of Line Integrals

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