Difference between revisions of "Conservative Vector Fields"

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===Path Independence===
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If a vector field '''F''' is the gradient of a scalar field ''G'' (i.e. if '''F''' is conservative), that is,
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:<math>\mathbf{F} = \nabla G ,</math>
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then by the multivariable chain rule, the derivative of the composition of ''G'' and '''r'''(''t'') is
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:<math>\frac{dG(\mathbf{r}(t))}{dt} = \nabla G(\mathbf{r}(t)) \cdot \mathbf{r}'(t) = \mathbf{F}(\mathbf{r}(t)) \cdot \mathbf{r}'(t)</math>
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which happens to be the integrand for the line integral of '''F''' on '''r'''(''t''). It follows, given a path ''C '', that
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:<math>\int_C \mathbf{F}(\mathbf{r})\cdot\,d\mathbf{r} = \int_a^b \mathbf{F}(\mathbf{r}(t))\cdot\mathbf{r}'(t)\,dt = \int_a^b \frac{dG(\mathbf{r}(t))}{dt}\,dt = G(\mathbf{r}(b)) - G(\mathbf{r}(a)).</math>
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In other words, the integral of '''F''' over ''C'' depends solely on the values of ''G'' at the points '''r'''(''b'') and '''r'''(''a''), and is thus independent of the path between them.  For this reason, a line integral of a conservative vector field is called ''path independent''.
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=== Stokes' Theorem ===
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Stokes' Theorem is effectively a generalization of Green's theorem to 3 dimensions, and the "curl" is a generalization of the quantity <math>\frac{\partial F_y}{\partial x} - \frac{\partial F_x}{\partial y}</math> to 3 dimensions. An arbitrary oriented surface <math>\sigma</math> can be articulated into a family of infinitesimal surfaces, some parallel to the xy-plane, others parallel to the zx-plane, and the remainder parallel to the yz-plane. Let <math>\mathbf{F}</math> denote an arbitrary vector field.
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Let <math>\sigma</math> be a surface that is parallel to the yz-plane with counter-clockwise oriented boundary <math>C</math>. Green's theorem gives:
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<math>\int_{\mathbf{q} \in C} \mathbf{F}(\mathbf{q}) \cdot d\mathbf{q} =
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\iint_{\mathbf{q} \in \sigma}\left(\frac{\partial F_z}{\partial y} - \frac{\partial F_y}{\partial z}\right)(\mathbf{dS} \cdot \mathbf{i}) =
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\iint_{\mathbf{q} \in \sigma}\left(\left(\frac{\partial F_z}{\partial y} - \frac{\partial F_y}{\partial z}\right)\mathbf{i}\right) \cdot \mathbf{dS}</math>
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<math>\mathbf{dS} \cdot \mathbf{i}</math> is positive if the normal direction to <math>\sigma</math> points in the positive x direction and is negative if otherwise. If the normal direction to <math>\sigma</math> points in the negative x direction, then <math>C</math> is oriented clockwise instead of counter-clockwise in the yz-plane.
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[[File:3D loop decomposition.svg|thumb|Decomposing a 3D loop into an ensemble of infinitesimal loops that are parallel to the yz, zx, or xy planes.]]
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Repeating this argument for <math>\sigma</math> being parallel to the zx-plane and xy-plane respectively gives:
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<math>\int_{\mathbf{q} \in C} \mathbf{F}(\mathbf{q}) \cdot d\mathbf{q} =
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\iint_{\mathbf{q} \in \sigma}\left(\frac{\partial F_x}{\partial z} - \frac{\partial F_z}{\partial x}\right)(\mathbf{dS} \cdot \mathbf{j}) =
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\iint_{\mathbf{q} \in \sigma}\left(\left(\frac{\partial F_x}{\partial z} - \frac{\partial F_z}{\partial x}\right)\mathbf{j}\right) \cdot \mathbf{dS}</math>
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and
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<math>\int_{\mathbf{q} \in C} \mathbf{F}(\mathbf{q}) \cdot d\mathbf{q} =
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\iint_{\mathbf{q} \in \sigma}\left(\frac{\partial F_y}{\partial x} - \frac{\partial F_x}{\partial y}\right)(\mathbf{dS} \cdot \mathbf{k}) =
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\iint_{\mathbf{q} \in \sigma}\left(\left(\frac{\partial F_y}{\partial x} - \frac{\partial F_x}{\partial y}\right)\mathbf{k}\right) \cdot \mathbf{dS}</math>
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Treating <math>\sigma</math> as an ensemble of infinitesimal surfaces parallel to the yz-plane, zx-plane, or xy-plane gives:
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<math>\int_{\mathbf{q} \in C} \mathbf{F}(\mathbf{q}) \cdot d\mathbf{q} =
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\iint_{\mathbf{q} \in \sigma}\left(
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\left(\frac{\partial F_z}{\partial y} - \frac{\partial F_y}{\partial z}\right)\mathbf{i} +
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\left(\frac{\partial F_x}{\partial z} - \frac{\partial F_z}{\partial x}\right)\mathbf{j} +
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\left(\frac{\partial F_y}{\partial x} - \frac{\partial F_x}{\partial y}\right)\mathbf{k}
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\right) \cdot \mathbf{dS} </math>
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This is Stokes' theorem, and
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<math>\nabla \times \mathbf{F} =
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\left(\frac{\partial F_z}{\partial y} - \frac{\partial F_y}{\partial z}\right)\mathbf{i} +
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\left(\frac{\partial F_x}{\partial z} - \frac{\partial F_z}{\partial x}\right)\mathbf{j} +
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\left(\frac{\partial F_y}{\partial x} - \frac{\partial F_x}{\partial y}\right)\mathbf{k}</math>
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is the "curl" of <math>\mathbf{F}</math> which generalizes the "circulation density" to 3 dimensions.
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The direction of <math>\nabla \times \mathbf{F}</math> at <math>\mathbf{q}</math> is effectively an "axis of rotation" around which the counterclockwise circulation density in a plane whose normal is parallel to <math>\nabla \times \mathbf{F}</math> is <math>|\nabla \times \mathbf{F}|</math>. Out of all planes that pass through <math>\mathbf{q}</math>, the plane whose normal is parallel to <math>\nabla \times \mathbf{F}</math> has the largest counterclockwise circulation density at <math>\mathbf{q}</math> which is <math>|\nabla \times \mathbf{F}|</math>.
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An arbitrary vector field <math>\mathbf{F}</math> that is differentiable everywhere is considered to be "irrotational" or "conservative" if <math>\nabla \times \mathbf{F} = \mathbf{0}</math> everywhere, or equivalently that <math>\int_{\mathbf{q} \in C} \mathbf{F}(\mathbf{q}) \cdot d\mathbf{q} = 0</math> for all continuous closed curves <math>C</math>
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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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===Scalar Potential===
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If '''F''' is a conservative vector field (also called ''irrotational'', ''curl-free'', or ''potential''), and its components have continuous partial derivatives, the '''potential''' of '''F''' with respect to a reference point <math>\mathbf r_0</math> is defined in terms of the line integral:
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:<math>V(\mathbf r) = -\int_C \mathbf{F}(\mathbf{r})\cdot\,d\mathbf{r} = -\int_a^b \mathbf{F}(\mathbf{r}(t))\cdot\mathbf{r}'(t)\,dt,</math>
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where ''C'' is a parametrized path from <math>\mathbf r_0</math> to <math>\mathbf r,</math>
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: <math>\mathbf{r}(t), a\leq t\leq b, \mathbf{r}(a)=\mathbf{r_0}, \mathbf{r}(b)=\mathbf{r}.</math>
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The fact that the line integral depends on the path ''C'' only through its terminal points <math>\mathbf r_0</math> and <math>\mathbf r</math> is, in essence, the '''path independence property''' of a conservative vector field. The fundamental theorem of line integrals implies that if ''V'' is defined in this way, then <math> \mathbf{F}= -\nabla V,</math> so that ''V'' is a scalar potential of the conservative vector field '''F'''. Scalar potential is not determined by the vector field alone: indeed, the gradient of a function is unaffected if a constant is added to it. If ''V'' is defined in terms of the line integral, the ambiguity of ''V'' reflects the freedom in the choice of the reference point <math>\mathbf r_0.</math>
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==Resources==
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* [https://en.wikibooks.org/wiki/Calculus/Vector_calculus Vector Calculus], Wikibooks: Calculus
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<strong>Conservative Vector Fields</strong>
 
<strong>Conservative Vector Fields</strong>
  
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* [https://www.youtube.com/watch?v=xVhHow_usMQ The Fundamental Theorem for Line Integrals] Video by Patrick JMT
 
* [https://www.youtube.com/watch?v=xVhHow_usMQ The Fundamental Theorem for Line Integrals] Video by Patrick JMT
 
* [https://www.youtube.com/watch?v=hT3YSXQaUHQ Potential Function of a Conservative Vector Field to Evaluate a Line Integral] Video by Krista King
 
* [https://www.youtube.com/watch?v=hT3YSXQaUHQ Potential Function of a Conservative Vector Field to Evaluate a Line Integral] Video by Krista King
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==Licensing==
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Content obtained and/or adapted from:
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* [https://en.wikibooks.org/wiki/Calculus/Vector_calculus#Volume,_path,_and_surface_integrals Vector calculus, Wikibooks: Calculus] under a CC BY-SA license
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* [https://en.wikipedia.org/wiki/Scalar_potential Scalar potential, Wikipedia] under a CC BY-SA license

Latest revision as of 08:15, 3 November 2021

Path Independence

If a vector field F is the gradient of a scalar field G (i.e. if F is conservative), that is,

then by the multivariable chain rule, the derivative of the composition of G and r(t) is

which happens to be the integrand for the line integral of F on r(t). It follows, given a path C , that

In other words, the integral of F over C depends solely on the values of G at the points r(b) and r(a), and is thus independent of the path between them. For this reason, a line integral of a conservative vector field is called path independent.

Stokes' Theorem

Stokes' Theorem is effectively a generalization of Green's theorem to 3 dimensions, and the "curl" is a generalization of the quantity to 3 dimensions. An arbitrary oriented surface can be articulated into a family of infinitesimal surfaces, some parallel to the xy-plane, others parallel to the zx-plane, and the remainder parallel to the yz-plane. Let denote an arbitrary vector field.

Let be a surface that is parallel to the yz-plane with counter-clockwise oriented boundary . Green's theorem gives:

is positive if the normal direction to points in the positive x direction and is negative if otherwise. If the normal direction to points in the negative x direction, then is oriented clockwise instead of counter-clockwise in the yz-plane.

Decomposing a 3D loop into an ensemble of infinitesimal loops that are parallel to the yz, zx, or xy planes.

Repeating this argument for being parallel to the zx-plane and xy-plane respectively gives:

and

Treating as an ensemble of infinitesimal surfaces parallel to the yz-plane, zx-plane, or xy-plane gives:

This is Stokes' theorem, 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 \nabla \times \mathbf{F} = \left(\frac{\partial F_z}{\partial y} - \frac{\partial F_y}{\partial z}\right)\mathbf{i} + \left(\frac{\partial F_x}{\partial z} - \frac{\partial F_z}{\partial x}\right)\mathbf{j} + \left(\frac{\partial F_y}{\partial x} - \frac{\partial F_x}{\partial y}\right)\mathbf{k}} is the "curl" 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}} which generalizes the "circulation density" to 3 dimensions.

The direction of 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 \mathbf{q}} is effectively an "axis of rotation" around which the counterclockwise circulation density in a plane whose normal is parallel to 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 \nabla \times \mathbf{F}} is . Out of all planes that pass through 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}} , the plane whose normal is parallel to 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 \nabla \times \mathbf{F}} has the largest counterclockwise circulation density at which is 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 |\nabla \times \mathbf{F}|} .

An arbitrary 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}} that is differentiable everywhere is considered to be "irrotational" or "conservative" if everywhere, or equivalently 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} \mathbf{F}(\mathbf{q}) \cdot d\mathbf{q} = 0} for all continuous closed curves 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}

Conservative vector fields

A vector field 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 \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 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 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 . 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 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 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 , 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 . Let 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 \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}}

Scalar Potential

If F is a conservative vector field (also called irrotational, curl-free, or potential), and its components have continuous partial derivatives, the potential of F with respect to a reference 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 r_0} is defined in terms of the line integral:

where C is a parametrized path from 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 r_0} to 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 r,}

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{r}(t), a\leq t\leq b, \mathbf{r}(a)=\mathbf{r_0}, \mathbf{r}(b)=\mathbf{r}.}

The fact that the line integral depends on the path C only through its terminal points 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 r_0} and is, in essence, the path independence property of a conservative vector field. The fundamental theorem of line integrals implies that if V is defined in this way, 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 \mathbf{F}= -\nabla V,} so that V is a scalar potential of the conservative vector field F. Scalar potential is not determined by the vector field alone: indeed, the gradient of a function is unaffected if a constant is added to it. If V is defined in terms of the line integral, the ambiguity of V reflects the freedom in the choice of the reference 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 r_0.}

Resources

Conservative Vector Fields


Finding a Potential Function of a Conservative Vector Field


The Fundamental Theorem of Line Integrals

Licensing

Content obtained and/or adapted from:

Retrieved from "https://mathresearch.utsa.edu/wiki/index.php?title=Conservative_Vector_Fields&oldid=3392"