Vector field - Revision history

DavidB4-bot: /* More formal treatment */Category

2016-07-29T20:18:24Z

More formal treatment: Category

DavidB4-bot: /* More formal treatment */clean up & uniformity

2016-07-13T21:20:02Z

More formal treatment: clean up & uniformity

SamHB: /* More formal treatment */ How come I can't find things? Oh, right. Because they're not in the right categories.

2015-06-24T17:28:27Z

More formal treatment: How come I can't find things? Oh, right. Because they're not in the right categories.

JLauttamus: remove {{jargon}} tag

2010-02-07T19:49:32Z

remove {{jargon}} tag

DanielPulido at 17:01, February 7, 2010

2010-02-07T17:01:52Z

SamHB: I think the set-theoretic stuff at the end is still too esoteric for our puposes, but I'm not going to take it out.

2009-12-28T03:21:11Z

I think the set-theoretic stuff at the end is still too esoteric for our puposes, but I'm not going to take it out.

Aschlafly: break out set theory formulation

2009-12-26T16:57:31Z

break out set theory formulation

FrankC: stressing a point of dimensions, trying to elaborate

2009-12-26T16:48:28Z

stressing a point of dimensions, trying to elaborate

Aschlafly: concise and clear

2009-12-26T16:32:39Z

concise and clear

New page

A '''vector field''' is a [[vector]] ''function'' that assigns a vector to each point in a region.

This is contrasted with a [[scalar field]] that merely assigns a scalar value to each point.

[[Category:vector analysis]]
[[Category:calculus]]
[[Category:mathematics]]

@@ Line 15: / Line 15: @@
 [[Category:Mathematics]]
 [[Category:Algebra]]
-[[Category:Vector analysis]]
+[[Category:Vector Analysis]]
 [[Category:Calculus]]
 [[Category:Physics]]

@@ Line 15: / Line 15: @@
 [[Category:Mathematics]]
 [[Category:Algebra]]
-[[Category:vector analysis]]
+[[Category:Vector analysis]]
-[[Category:calculus]]
+[[Category:Calculus]]
 [[Category:Physics]]

← Older revision		Revision as of 19:49, February 7, 2010
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−	~~{{jargon}}~~
−
	A '''vector field''' is an assignment of a [[vector]] to each point in some region of space. A simple example of a vector field is the wind velocity (a vector) at each point on the Earth's surface. Vector fields are extremely important in science and engineering—some examples are the electric and magnetic fields of [[Maxwell's Equations\|electrodynamics]], and the gravitational field of [[gravitation\|Newtonian gravity]].		A '''vector field''' is an assignment of a [[vector]] to each point in some region of space. A simple example of a vector field is the wind velocity (a vector) at each point on the Earth's surface. Vector fields are extremely important in science and engineering—some examples are the electric and magnetic fields of [[Maxwell's Equations\|electrodynamics]], and the gravitational field of [[gravitation\|Newtonian gravity]].

← Older revision		Revision as of 17:01, February 7, 2010
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		+	{{jargon}}
		+
	A '''vector field''' is an assignment of a [[vector]] to each point in some region of space. A simple example of a vector field is the wind velocity (a vector) at each point on the Earth's surface. Vector fields are extremely important in science and engineering—some examples are the electric and magnetic fields of [[Maxwell's Equations\|electrodynamics]], and the gravitational field of [[gravitation\|Newtonian gravity]].		A '''vector field''' is an assignment of a [[vector]] to each point in some region of space. A simple example of a vector field is the wind velocity (a vector) at each point on the Earth's surface. Vector fields are extremely important in science and engineering—some examples are the electric and magnetic fields of [[Maxwell's Equations\|electrodynamics]], and the gravitational field of [[gravitation\|Newtonian gravity]].

@@ Line 1: / Line 1: @@
-A '''vector field''' is a [[vector]] ''function'' that assigns a vector to each point in a region.
+A '''vector field''' is an assignment of a [[vector]] to each point in some region of space.  A simple example of a vector field is the wind velocity (a vector) at each point on the Earth's surface.  Vector fields are extremely important in science and engineering&mdash;some examples are the electric and magnetic fields of [[Maxwell's Equations|electrodynamics]], and the gravitational field of [[gravitation|Newtonian gravity]].
-This is contrasted with a [[scalar field]] that merely assigns a scalar value to each point.
+In addition to the usual operations of vectors (addition, scalar multiplication, [[dot product]], and [[cross product]]), vector fields have some differentiation operations that compare the values of the field at different points.  The common operations are the [[divergence]] and the [[curl]].
-== Formulation in Set Theory ==
+In analogy with a vector field, a [[scalar field]] is an assignment of a plain number ("scalar") to each point.  There are tensor fields as well, assigning a [[tensor]] to every point in space.
 If <math>S</math> is a subset of the <math>\mathbb{R}^n</math>, then a vector field can be seen as the function <math>\vec{V}: S \rightarrow \mathbb{R}^n</math>, which maps an n-dimensional positional vector <math>\vec{x}=(x_1, x_2, \dots, x_n) \in \mathbb{R}^n</math> to each point <math>\vec{p} \in S</math>:

← Older revision		Revision as of 16:48, December 26, 2009
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−	A '''vector field''' is a [[vector]] ''function'' that assigns a vector to each point in a region.	+	A '''vector field''' is a [[vector]] ''function'' that assigns a vector to each point in a region. So, if <math>S</math> is a subset of the <math>\mathbb{R}^n</math>, then a vector field can be seen as the function <math>\vec{V}: S \rightarrow \mathbb{R}^n</math>, which maps an n-dimensional vector <math>\vec{x}=(x_1, x_2, \dots, x_n) \in \mathbb{R}^n</math> to each point <math>\vec{p} \in S</math>:
		+
		+	<center><math>\vec{V}(\vec{p}) = \vec{x}</math></center>
		+

	This is contrasted with a [[scalar field]] that merely assigns a scalar value to each point.		This is contrasted with a [[scalar field]] that merely assigns a scalar value to each point.