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| [[File:Normal and tangent illustration.png|right|thumb|Illustration of tangential and normal components of a vector to a surface.]]
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| In [[mathematics]], given a [[Vector (geometric)|vector]] at a point on a [[curve]], that vector can be decomposed uniquely as a sum of two vectors, one [[tangent]] to the curve, called the '''tangential component''' of the vector, and another one [[perpendicular]] to the curve, called the '''normal component''' of the vector. Similarly a vector at a point on a [[surface]] can be broken down the same way.
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| More generally, given a [[submanifold]] ''N'' of a [[manifold]] ''M'', and a vector in the [[tangent space]] to ''M'' at a point of ''N'', it can be decomposed into the component tangent to ''N'' and the component normal to ''N''.
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| ==Formal definition==
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| ===Surface===
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| More formally, let <math>S</math> be a surface, and <math>x</math> be a point on the surface. Let <math>\mathbf{v}</math> be a vector at <math>x.</math> Then one can write uniquely <math>\mathbf{v}</math> as a sum
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| : <math>\mathbf{v}=\mathbf{v}_{\parallel}+\mathbf{v}_\perp</math>
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| where the first vector in the sum is the tangential component and the second one is the normal component. It follows immediately that these two vectors are perpendicular to each other.
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| To calculate the tangential and normal components, consider a [[surface normal|unit normal]] to the surface, that is, a [[unit vector]] <math>\hat{n}</math> perpendicular to <math>S</math> at <math>x.</math> Then,
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| :<math>\mathbf{v}_\perp = (\mathbf{v}\cdot\hat{n})\hat{n}</math>
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| and thus
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| :<math>\mathbf{v}_\parallel = \mathbf{v} - \mathbf{v}_\perp</math>
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| where "<math>\cdot</math>" denotes the [[dot product]]. Another formula for the tangential component is
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| :<math>\mathbf{v}_\parallel = -\hat{n}\times(\hat{n}\times\mathbf{v}),</math>
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| where "<math>\times</math>" denotes the [[cross product]].
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| Note that these formulas do not depend on the particular unit normal <math>\hat{n}</math> used (there exist two unit normals to any surface at a given point, pointing in opposite directions, so one of the unit normals is the negative of the other one).
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| ===Submanifold===
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| More generally, given a [[submanifold]] ''N'' of a [[manifold]] ''M'' and
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| a point <math>p \in N</math>, we get a [[short exact sequence]]
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| involving the [[tangent space]]s:
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| :<math>T_p N \to T_p M \to T_p M / T_p N</math>
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| The [[quotient space]] <math>T_p M / T_p N</math> is a generalized space of normal vectors.
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| If ''M'' is a [[Riemannian manifold]], the above sequence splits, and the tangent space of ''M'' at ''p'' decomposes as a [[direct sum of vector spaces|direct sum]] of the component tangent to ''N'' and the component normal to ''N'':
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| :<math>T_p M = T_p N \oplus N_p N := (T_p N)^\perp</math>
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| Thus every [[tangent vector]] <math>v \in T_p M</math> splits as
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| <math>v = v_\parallel + v_\perp</math>,
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| where <math>v_\parallel \in T_p N</math> and <math>v_\perp \in N_p N := (T_p N)^\perp</math>. | |
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| ==Computations==
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| Suppose ''N'' is given by non-degenerate equations.
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| If ''N'' is given explicitly, via [[parametric equation]]s (such as a [[parametric curve]]), then the derivative gives a spanning set for the tangent bundle (it's a basis if and only if the parametrization is an [[immersion (mathematics)|immersion]]).
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| If ''N'' is given [[implicit surface|implicitly]] (as in the above description of a surface, or more generally as a [[hypersurface]]) as a [[level set]] or intersection of level surfaces for <math>g_i</math>, then the gradients of <math>g_i</math> span the normal space.
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| In both cases, we can again compute using the dot product; the cross product is special to 3 dimensions though.
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| ==Applications==
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| * [[Lagrange multipliers]] : constrained [[Critical point (mathematics)|critical points]] are where the tangential component of the [[total derivative]] vanish.
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| * [[Surface normal]]
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| ==References==
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| *{{cite book
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| | last = Rojansky
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| | first = Vladimir
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| | title = Electromagnetic fields and waves
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| | publisher = New York: Dover Publications
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| | date = 1979
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| | pages =
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| | isbn = 0-486-63834-0
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| }}
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| * Benjamin Crowell (2003) ''Newtonian physics.'' ([http://www.faqs.org/docs/Newtonian/Newtonian_179.htm online version]) ISBN 0-9704670-1-X.
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| [[Category:Differential geometry]]
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Hi there, I am Alyson Pomerleau and I believe it sounds quite great when you say it. My spouse doesn't like it the way I do but what I truly like doing is caving but I don't have the time recently. Mississippi is where her home is but her husband wants them to transfer. Invoicing is what I do.
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