Conservative Vector Fields - Revision history

Lila: /* Scalar Potential Function */

2021-11-03T14:15:55Z

Scalar Potential Function

Lila at 14:14, 3 November 2021

2021-11-03T14:14:36Z

Lila at 14:10, 3 November 2021

2021-11-03T14:10:17Z

Lila at 02:31, 11 October 2021

2021-10-11T02:31:48Z

Lila: /* Resources */

2021-10-11T02:28:52Z

Resources

Lila at 02:28, 11 October 2021

2021-10-11T02:28:16Z

Glenn.lahodny at 11:57, 5 May 2021

2021-05-05T11:57:33Z

Glenn.lahodny: Created page with "Conservative Vector Fields * [https://www.youtube.com/watch?v=cVTcsHcispQ Conservative Vector Fields] Video by James Sousa, Math is Power 4U * [https://www.y..."

2021-05-05T11:57:05Z

Created page with "Conservative Vector Fields * [https://www.youtube.com/watch?v=cVTcsHcispQ Conservative Vector Fields] Video by James Sousa, Math is Power 4U * [https://www.y..."

New page

Conservative Vector Fields

* [https://www.youtube.com/watch?v=cVTcsHcispQ Conservative Vector Fields] Video by James Sousa, Math is Power 4U
* [https://www.youtube.com/watch?v=VLTUku45Qs4 Conservative Vector Fields - The Definition and a Few Remarks] Video by Patrick JMT
* [https://www.youtube.com/watch?v=gAb1ZTD41wo Showing a Vector Field on R^2 is Conservative] Video by Patrick JMT

Finding a Potential Function of a Conservative Vector Field
* [https://www.youtube.com/watch?v=nQkHh2psLck Determining the Potential Function of a Conservative Vector Field] Video by James Sousa, Math is Power 4U
* [https://www.youtube.com/watch?v=iLAK2IsQ_Uo Finding a Potential for a Conservative Vector Field] Video by Patrick JMT
* [https://www.youtube.com/watch?v=tslJEOnt9aY Finding a Potential for a Conservative Vector Field Ex 2] Video by Patrick JMT
* [https://www.youtube.com/watch?v=nY4mW_R-T40 Potential Function of a Conservative Vector Field] Video by Krista King
* [https://www.youtube.com/watch?v=Rb12YouJXyA Potential Function of a Conservative Vector Field in 3D] Video by Krista King

The Fundamental Theorem of Line Integrals
* [https://www.youtube.com/watch?v=Td6g2bvJh18 The Fundamental Theorem of Line Integrals Part 1] Video by James Sousa, Math is Power 4U
* [https://www.youtube.com/watch?v=I8BtXV-xgxM The Fundamental Theorem of Line Integrals Part 2] Video by James Sousa, Math is Power 4U
* [https://www.youtube.com/watch?v=00RN-hw9BXc The Fundamental Theorem of Line Integrals on a Closed Path] Video by James Sousa, Math is Power 4U
* [https://www.youtube.com/watch?v=62oBGKSjYiY Ex 1: Fundamental Theorem of Line Integrals in the Plane] Video by James Sousa, Math is Power 4U
* [https://www.youtube.com/watch?v=Eh6OinLiGWg Ex 2: Fundamental Theorem of Line Integrals in the Plane] Video by James Sousa, Math is Power 4U
* [https://www.youtube.com/watch?v=pbHlZR2TD7M Ex 3: Fundamental Theorem of Line Integrals in the Plane] Video by James Sousa, Math is Power 4U
* [https://www.youtube.com/watch?v=e2g7321ne48 Ex 4: Fundamental Theorem of Line Integrals in Space] Video by James Sousa, Math is Power 4U
* [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

@@ Line 72: / Line 72: @@
 <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>
-==Scalar Potential Function==
+===Scalar Potential===
-If '''F''' is a [[conservative vector field]] (also called ''irrotational'', ''[[Curl (mathematics)|curl]]-free'', or ''potential''), and its components have [[continuous function|continuous]] [[partial derivative]]s, the '''potential''' of '''F''' with respect to a reference point <math>\mathbf r_0</math> is defined in terms of the [[line integral]]:
+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:
 :<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>
@@ Line 81: / Line 81: @@
 : <math>\mathbf{r}(t), a\leq t\leq b, \mathbf{r}(a)=\mathbf{r_0}, \mathbf{r}(b)=\mathbf{r}.</math>
-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 [[Gradient theorem|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>
+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>
 ==Resources==

@@ Line 71: / Line 71: @@
 <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>
@@ Line 105: / Line 117: @@
 Content obtained and/or adapted from:
 * [https://en.wikibooks.org/wiki/Calculus/Vector_calculus#Volume,_path,_and_surface_integrals Vector calculus, Wikibooks: Calculus] under a CC BY-SA license

@@ Line 1: / Line 1: @@
 === Stokes' Theorem ===
@@ Line 86: / Line 101: @@
 * [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

← Older revision		Revision as of 02:31, 11 October 2021
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		+	=== Stokes' Theorem ===
		+
		+	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.
		+
		+	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:
		+
		+	<math>\int_{\mathbf{q} \in C} \mathbf{F}(\mathbf{q}) \cdot d\mathbf{q} =
		+	\iint_{\mathbf{q} \in \sigma}\left(\frac{\partial F_z}{\partial y} - \frac{\partial F_y}{\partial z}\right)(\mathbf{dS} \cdot \mathbf{i}) =
		+	\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>
		+
		+	<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.
		+
		+	[[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.]]
		+
		+	Repeating this argument for <math>\sigma</math> being parallel to the zx-plane and xy-plane respectively gives:
		+
		+	<math>\int_{\mathbf{q} \in C} \mathbf{F}(\mathbf{q}) \cdot d\mathbf{q} =
		+	\iint_{\mathbf{q} \in \sigma}\left(\frac{\partial F_x}{\partial z} - \frac{\partial F_z}{\partial x}\right)(\mathbf{dS} \cdot \mathbf{j}) =
		+	\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>
		+
		+	and
		+
		+	<math>\int_{\mathbf{q} \in C} \mathbf{F}(\mathbf{q}) \cdot d\mathbf{q} =
		+	\iint_{\mathbf{q} \in \sigma}\left(\frac{\partial F_y}{\partial x} - \frac{\partial F_x}{\partial y}\right)(\mathbf{dS} \cdot \mathbf{k}) =
		+	\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>
		+
		+	Treating <math>\sigma</math> as an ensemble of infinitesimal surfaces parallel to the yz-plane, zx-plane, or xy-plane gives:
		+
		+	<math>\int_{\mathbf{q} \in C} \mathbf{F}(\mathbf{q}) \cdot d\mathbf{q} =
		+	\iint_{\mathbf{q} \in \sigma}\left(
		+	\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}
		+	\right) \cdot \mathbf{dS} </math>
		+
		+	This is Stokes' theorem, and
		+	<math>\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}</math>
		+	is the "curl" of <math>\mathbf{F}</math> which generalizes the "circulation density" to 3 dimensions.
		+
		+	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>.
		+
		+	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>
		+
	=== Conservative vector fields ===		=== Conservative vector fields ===

← Older revision		Revision as of 02:28, 11 October 2021
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	==Resources==		==Resources==
		+	* [https://en.wikibooks.org/wiki/Calculus/Vector_calculus Vector Calculus], Wikibooks: Calculus
		+
	<strong>Conservative Vector Fields</strong>		<strong>Conservative Vector Fields</strong>

@@ Line 4: / Line 4: @@
 * [https://www.youtube.com/watch?v=VLTUku45Qs4 Conservative Vector Fields - The Definition and a Few Remarks] Video by Patrick JMT
 * [https://www.youtube.com/watch?v=gAb1ZTD41wo Showing a Vector Field on R^2 is Conservative] Video by Patrick JMT
 <strong>Finding a Potential Function of a Conservative Vector Field</strong>
@@ Line 11: / Line 12: @@
 * [https://www.youtube.com/watch?v=nY4mW_R-T40 Potential Function of a Conservative Vector Field] Video by Krista King
 * [https://www.youtube.com/watch?v=Rb12YouJXyA Potential Function of a Conservative Vector Field in 3D] Video by Krista King
 <strong>The Fundamental Theorem of Line Integrals</strong>