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((More citations needed|date=December 2023))
In [[3-dimensional space|3-dimensional]] [[geometry]] and [[vector calculus]], an '''area vector''' is a [[Euclidean vector|vector]] combining an [[Area|area quantity]] with a [[direction (geometry)|direction]], thus representing an '''oriented area''' in three dimensions. 

Every [[Bounded set|bounded]] [[Surface (topology)|surface]] in three dimensions can be associated with a unique area vector called its '''vector area'''. It is equal to the [[surface integral]] of the [[surface normal]], and distinct from the usual ([[Scalar (mathematics)|scalar]]) [[surface area]]. 

Vector area can be seen as the three dimensional generalization of [[signed area]] in two dimensions.

== Definition ==
For a finite planar surface of scalar area ((mvar|S)) and [[unit normal]] ((math|'''n̂''')), the vector area ((math|'''S''')) is defined as the unit normal scaled by the area:
<math display="block">\mathbf{S} = \mathbf{\hat n}S</math>

For an [[orientable]] surface ((mvar|S)) composed of a set ((mvar|S<sub>i</sub>)) of flat [[Facet_(geometry)|facet]] areas, the vector area of the surface is given by
<math display="block">\mathbf{S} = \sum_i \mathbf{\hat n}_i S_i</math>
where ((math|'''n̂'''<sub>''i''</sub>)) is the unit normal vector to the area ((mvar|S<sub>i</sub>)).

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.
<math display="block">d\mathbf{S} = \mathbf{\hat n}dS</math>
where ((math|'''n̂''')) is the local unit vector perpendicular to ((mvar|dS)). Integrating gives the vector area for the surface.
<math display="block">\mathbf{S} = \int d\mathbf{S}</math>

== Properties ==
The vector area of a surface can be interpreted as the (signed) projected area or "shadow" of the surface in the plane in which it is greatest; its direction is given by that plane's normal. 

For a curved or faceted (i.e. non-planar) surface, the vector area is smaller in magnitude than the actual [[surface area]]. As an extreme example, a [[closed surface]] can possess arbitrarily large area, but its vector area is necessarily zero.<ref>((cite book| first=Murray R.|last=Spiegel|title=Theory and problems of vector analysis|series=Schaum's Outline Series|publisher=McGraw Hill| date=1959|page=25))</ref> 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 vector area of a [[parallelogram]] is given by the [[cross product]] of the two vectors that span it; it is twice the (vector) area of the triangle formed by the same vectors. In general, the vector area of any surface whose boundary consists of a sequence of straight [[Line segment|line segments]] (analogous to a [[polygon]] in two dimensions) can be calculated using a series of cross products corresponding to a [[Triangle mesh|triangularization]] of the surface. This is the generalization of the [[Shoelace formula]] to three dimensions.

Using [[Stokes' theorem]] applied to an appropriately chosen vector field, a boundary integral for the vector area can be derived:
<math display="block">\mathbf{S} = \frac{1}{2} \oint_{\partial S} \mathbf r \times d \mathbf r</math>
where <math>\partial S</math> is the boundary of ((mvar|S)), i.e. one or more oriented closed space [[Curve|curves]]. This is analogous to the two dimensional [[Green's theorem#Area calculation|area calculation using Green's theorem]].

== Applications ==
Area vectors are used when calculating [[Surface integral|surface integrals]], such as when determining the [[flux]] of a [[vector field]] through a surface. The flux is given by the integral of the [[dot product]] of the field and the (infinitesimal) area vector. When the field is constant over the surface the integral simplifies to the dot product of the field and the vector area of the surface.

=== Projection of area onto planes ===
The [[projected area]] onto a plane is given by the [[dot product]] of the vector area '''S''' and the target plane unit normal ((math|'''m̂''')):
<math display="block">A_{\parallel} = \mathbf{S} \cdot \hat \mathbf m</math>
For example, the projected area onto the ((mvar|xy))-plane is equivalent to the ((mvar|z))-component of the vector area, and is also equal to
<math display="block">\mathbf{S}_z = \left| \mathbf{S} \right| \cos \theta</math>
where ((mvar|θ)) is the angle between the plane normal ((math|'''n̂''')) and the ((mvar|z))-axis.

== See also ==
* [[Bivector]], representing an oriented area in any number of dimensions
* [[De Gua's theorem]], on the decomposition of vector area into orthogonal components
* [[Cross product]]
* [[Surface normal]]
* [[Surface integral]]

== Notes ==
<references />

[[Category:Area]]
[[Category:Vectors (mathematics and physics)]]
[[Category:Analytic geometry]]

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