World Library  
Flag as Inappropriate
Email this Article

Vector area

Article Id: WHEBN0003103326
Reproduction Date:

Title: Vector area  
Author: World Heritage Encyclopedia
Language: English
Subject: Flux, Defining equation (physics), Analytic geometry, Physical quantity, List of equations in fluid mechanics
Collection: Analytic Geometry, Area, Vectors (Mathematics and Physics)
Publisher: World Heritage Encyclopedia

Vector area

In 3-dimensional geometry, for a finite planar surface of scalar area S and unit normal \hat{n}, the vector area \mathbf{S} is defined as the unit normal scaled by the area:

\mathbf{S} = \mathbf{\hat{n}}S

For an orientable surface S composed of a set S_i of flat facet areas, the vector area of the surface is given by

\mathbf{S} = \sum_i \mathbf{\hat{n}}_i S_i

where \mathbf{\hat{n}}_i is the unit normal vector to the area S_i.

For bounded, oriented curved surfaces that are sufficiently well-behaved, we can still define vector area. First, we split the surface into infinitesimal elements, each of which is effectively flat. For each infinitesimal element of area, we have an area vector, also infinitesimal.

d\mathbf{S} = \mathbf{\hat{n}}dS

where \mathbf{\hat{n}} is the local unit vector perpendicular to dS. Integrating gives the vector area for the surface.

\mathbf{S} = \int d\mathbf{S}

For a curved or faceted surface, the vector area is smaller in magnitude than the area. As an extreme example, a closed surface can possess arbitrarily large area, but its vector area is necessarily zero.[1] Surfaces that share a boundary may have very different areas, but they must have the same vector area—the vector area is entirely determined by the boundary. These are consequences of Stokes theorem.

The concept of an area vector simplifies the equation for determining the flux through the surface. Consider a planar surface in a uniform field. The flux can be written as the dot product of the field and area vector. This is much simpler than multiplying the field strength by the surface area and the cosine of the angle between the field and the surface normal.

Projection of area onto planes

The projected area onto (for example) the x-y plane is equivalent to the z-component of the vector area, and is given by

\mathbf{S_z} = \left| \mathbf{S} \right| \cos \theta

where \theta is the angle between the plane normal and the z-axis.

See also


  1. ^ Murray R. Spiegel, Theory and problems of vector analysis, Schaum's Outline Series, McGraw Hill, 1959, p. 25.
This article was sourced from Creative Commons Attribution-ShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for and content contributors is made possible from the U.S. Congress, E-Government Act of 2002.
Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.
By using this site, you agree to the Terms of Use and Privacy Policy. World Heritage Encyclopedia™ is a registered trademark of the World Public Library Association, a non-profit organization.

Copyright © World Library Foundation. All rights reserved. eBooks from Project Gutenberg are sponsored by the World Library Foundation,
a 501c(4) Member's Support Non-Profit Organization, and is NOT affiliated with any governmental agency or department.