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Vector field (view source)

Revision as of 16:57, December 26, 2009

41 bytes added ,  16:57, December 26, 2009
break out set theory formulation
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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>:
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A '''vector field''' is a [[vector]] ''function'' that assigns a vector to each point in a region.  
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<center><math>\vec{V}(\vec{p}) = \vec{x}</math></center>
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This is contrasted with a [[scalar field]] that merely assigns a scalar value to each point.
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== Formulation in Set Theory ==
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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>:
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This is contrasted with a [[scalar field]] that merely assigns a scalar value to each point.
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<center><math>\vec{V}(\vec{p}) = \vec{x}</math></center>
    
[[Category:vector analysis]]
 
[[Category:vector analysis]]
 
[[Category:calculus]]
 
[[Category:calculus]]
 
[[Category:mathematics]]
 
[[Category:mathematics]]
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