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<title>Vector area</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Vector area</span></span>

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<p>In <a href="3-dimensional_space" class="mw-redirect" title="3-dimensional space">3-dimensional</a> <a href="Geometry" title="Geometry">geometry</a> and <a href="Vector_calculus" title="Vector calculus">vector calculus</a>, an <b>area vector</b> is a <a href="Euclidean_vector" title="Euclidean vector">vector</a> combining an <a href="Area" title="Area">area quantity</a> with a <a href="Direction_(geometry)" title="Direction (geometry)">direction</a>, thus representing an <b>oriented area</b> in three dimensions.

</p><p>Every <a href="Bounded_set" title="Bounded set">bounded</a> <a href="Surface_(topology)" title="Surface (topology)">surface</a> in three dimensions can be associated with a unique area vector called its <b>vector area</b>. It is equal to the <a href="Surface_integral" title="Surface integral">surface integral</a> of the <a href="Surface_normal" class="mw-redirect" title="Surface normal">surface normal</a>, and distinct from the usual (<a href="Scalar_(mathematics)" title="Scalar (mathematics)">scalar</a>) <a href="Surface_area" title="Surface area">surface area</a>.

</p><p>Vector area can be seen as the three dimensional generalization of <a href="Signed_area" title="Signed area">signed area</a> in two dimensions.

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<div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2></div>

<p>For a finite planar surface of scalar area <span class="texhtml mvar" style="font-style:italic;">S</span> and <a href="Unit_normal" class="mw-redirect" title="Unit normal">unit normal</a> <span class="texhtml"><span class="sfrac nowrap;"><span style="display:none; display:inline-block; text-align:center;"><span style="display:block; line-height:0.3; font-size:70%;">^</span><span style="display:block; line-height:0.3;"><b>n</b></span></span></span></span>, the vector area <span class="texhtml"><b>S</b></span> is defined as the unit normal scaled by the area:

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {S} ={\hat {\mathbf {n} }}S}</annotation>

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</p><p>For an <a href="Orientable" class="mw-redirect" title="Orientable">orientable</a> surface <span class="texhtml mvar" style="font-style:italic;">S</span> composed of a set <span class="texhtml mvar" style="font-style:italic;">S<sub>i</sub></span> of flat <a href="Facet_(geometry)" title="Facet (geometry)">facet</a> areas, the vector area of the surface is given by

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {S} =\sum _{i}{\hat {\mathbf {n} }}_{i}S_{i}}</annotation>

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where <span class="texhtml"><span class="sfrac nowrap;"><span style="display:none; display:inline-block; text-align:center;"><span style="display:block; line-height:0.3; font-size:70%;">^</span><span style="display:block; line-height:0.3;"><b>n</b></span></span></span><sub><i>i</i></sub></span> is the unit normal vector to the area <span class="texhtml mvar" style="font-style:italic;">S<sub>i</sub></span>.

</p><p>For bounded, oriented curved surfaces that are sufficiently <a href="Well-behaved" class="mw-redirect" title="Well-behaved">well-behaved</a>, 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.

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<annotation encoding="application/x-tex">{\displaystyle d\mathbf {S} ={\hat {\mathbf {n} }}\ dS}</annotation>

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where <span class="texhtml"><span class="sfrac nowrap;"><span style="display:none; display:inline-block; text-align:center;"><span style="display:block; line-height:0.3; font-size:70%;">^</span><span style="display:block; line-height:0.3;"><b>n</b></span></span></span></span> is the local unit vector perpendicular to <span class="texhtml mvar" style="font-style:italic;">dS</span>. Integrating gives the vector area for the surface.

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {S} =\int d\mathbf {S} }</annotation>

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<div class="mw-heading mw-heading2"><h2 id="Properties">Properties</h2></div>

<p>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.

</p><p>For a curved or faceted (i.e. non-planar) surface, the vector area is smaller in magnitude than the actual <a href="Surface_area" title="Surface area">surface area</a>. As an extreme example, a <a href="Closed_surface" class="mw-redirect" title="Closed surface">closed surface</a> can possess arbitrarily large area, but its vector area is necessarily zero.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> 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 <a href="Stokes'_theorem" title="Stokes' theorem">Stokes' theorem</a>.

</p><p>The vector area of a <a href="Parallelogram" title="Parallelogram">parallelogram</a> is given by the <a href="Cross_product" title="Cross product">cross product</a> 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 <a href="Line_segment" title="Line segment">line segments</a> (analogous to a <a href="Polygon" title="Polygon">polygon</a> in two dimensions) can be calculated using a series of cross products corresponding to a <a href="Triangle_mesh" title="Triangle mesh">triangularization</a> of the surface. This is the generalization of the <a href="Shoelace_formula" title="Shoelace formula">Shoelace formula</a> to three dimensions.

</p><p>Using <a href="Stokes'_theorem" title="Stokes' theorem">Stokes' theorem</a> applied to an appropriately chosen vector field, a boundary integral for the vector area can be derived:

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {S} ={\frac {1}{2}}\oint _{\partial S}\mathbf {r} \times d\mathbf {r} }</annotation>

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where <span class="texhtml">∂<i>S</i></span> is the boundary of <span class="texhtml"><i>S</i></span>, i.e. one or more oriented closed space <a href="Curve" title="Curve">curves</a>. This is analogous to the two dimensional <a href="Green's_theorem#Area_calculation" title="Green's theorem">area calculation using Green's theorem</a>.

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<div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2></div>

<p>Area vectors are used when calculating <a href="Surface_integral" title="Surface integral">surface integrals</a>, such as when determining the <a href="Flux" title="Flux">flux</a> of a <a href="Vector_field" title="Vector field">vector field</a> through a surface. The flux is given by the integral of the <a href="Dot_product" title="Dot product">dot product</a> 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.

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<div class="mw-heading mw-heading3"><h3 id="Projection_of_area_onto_planes">Projection of area onto planes</h3></div>

<p>The <a href="Projected_area" title="Projected area">projected area</a> onto a plane is given by the <a href="Dot_product" title="Dot product">dot product</a> of the vector area <span class="texhtml"><b>S</b></span> and the target plane unit normal <span class="texhtml"><span class="sfrac nowrap;"><span style="display:none; display:inline-block; text-align:center;"><span style="display:block; line-height:0.3; font-size:70%;">^</span><span style="display:block; line-height:0.3;"><b>m</b></span></span></span></span>:

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<annotation encoding="application/x-tex">{\displaystyle A_{\parallel }=\mathbf {S} \cdot {\hat {\mathbf {m} }}}</annotation>

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For example, the projected area onto the <span class="texhtml mvar" style="font-style:italic;">xy</span>-plane is equivalent to the <span class="texhtml mvar" style="font-style:italic;">z</span>-component of the vector area, and is also equal to

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {S} _{z}=\left|\mathbf {S} \right|\cos \theta }</annotation>

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where <span class="texhtml"><i>θ</i></span> is the angle between the plane normal <span class="texhtml"><span class="sfrac nowrap;"><span style="display:none; display:inline-block; text-align:center;"><span style="display:block; line-height:0.3; font-size:70%;">^</span><span style="display:block; line-height:0.3;"><b>n</b></span></span></span></span> and the <span class="texhtml mvar" style="font-style:italic;">z</span>-axis.

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<ul><li><a href="Bivector" title="Bivector">Bivector</a>, representing an oriented area in any number of dimensions</li>

<li><a href="De_Gua's_theorem" title="De Gua's theorem">De Gua's theorem</a>, on the decomposition of vector area into orthogonal components</li>

<li><a href="Cross_product" title="Cross product">Cross product</a></li>

<li><a href="Surface_normal" class="mw-redirect" title="Surface normal">Surface normal</a></li>

<li><a href="Surface_integral" title="Surface integral">Surface integral</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2></div>

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</style><cite id="CITEREFSpiegel1959" class="citation book cs1">Spiegel, Murray R. (1959). <i>Theory and problems of vector analysis</i>. Schaum's Outline Series. McGraw Hill. p. 25.</cite></span>

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