Conservative Vector Fields

Conservative vector fields

A 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}} for which $\nabla \times \mathbf {F} =\mathbf {0}$ at all points is an "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}} 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 $C_{2}$ 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 $-C_{2}$ 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 $\mathbf {q} _{1}$. 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 $\int _{\mathbf {q} \in C_{3}}\mathbf {F} (\mathbf {q} )\cdot d\mathbf {q} =\iint _{\mathbf {q} \in \sigma }(\nabla \times \mathbf {F} )|_{\mathbf {q} }\cdot \mathbf {dS} =0$. 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 $C_{2}$: 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:

$\int _{\mathbf {q} \in C_{3}}\mathbf {F} (\mathbf {q} )\cdot d\mathbf {q} =0$ 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

• Vector Calculus, Wikibooks: Calculus

Conservative Vector Fields

• Conservative Vector Fields Video by James Sousa, Math is Power 4U
• Conservative Vector Fields - The Definition and a Few Remarks Video by Patrick JMT
• Showing a Vector Field on R^2 is Conservative Video by Patrick JMT

Finding a Potential Function of a Conservative Vector Field

• Determining the Potential Function of a Conservative Vector Field Video by James Sousa, Math is Power 4U
• Finding a Potential for a Conservative Vector Field Video by Patrick JMT
• Finding a Potential for a Conservative Vector Field Ex 2 Video by Patrick JMT
• Potential Function of a Conservative Vector Field Video by Krista King
• Potential Function of a Conservative Vector Field in 3D Video by Krista King

The Fundamental Theorem of Line Integrals

• The Fundamental Theorem of Line Integrals Part 1 Video by James Sousa, Math is Power 4U
• The Fundamental Theorem of Line Integrals Part 2 Video by James Sousa, Math is Power 4U
• The Fundamental Theorem of Line Integrals on a Closed Path Video by James Sousa, Math is Power 4U
• Ex 1: Fundamental Theorem of Line Integrals in the Plane Video by James Sousa, Math is Power 4U
• Ex 2: Fundamental Theorem of Line Integrals in the Plane Video by James Sousa, Math is Power 4U
• Ex 3: Fundamental Theorem of Line Integrals in the Plane Video by James Sousa, Math is Power 4U
• Ex 4: Fundamental Theorem of Line Integrals in Space Video by James Sousa, Math is Power 4U
• The Fundamental Theorem for Line Integrals Video by Patrick JMT
• Potential Function of a Conservative Vector Field to Evaluate a Line Integral Video by Krista King