Tangential and normal components

 ( Differential Geometry ) 

In mathematics, given a 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.

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 More formally, let $S$ be a surface, and $x$ be a point on the surface. Let $\backslash mathbf\{v\}$ be a vector at Then one can write uniquely $\backslash mathbf\{v\}$ as a sum $$\backslash mathbf\{v\}\; =\; \backslash mathbf\{v\}\_\backslash parallel\; +\; \backslash mathbf\{v\}\_\backslash perp$$ 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.

To calculate the tangential and normal components, consider a surface normal to the surface, that is, a unit vector $\backslash hat\backslash mathbf\{n\}$ perpendicular to $S$ at Then, $$\backslash mathbf\{v\}\_\backslash perp\; =\; \backslash left(\backslash mathbf\{v\}\; \backslash cdot\; \backslash hat\backslash mathbf\{n\}\backslash right)\; \backslash hat\backslash mathbf\{n\}$$ and thus $$\backslash mathbf\{v\}\_\backslash parallel\; =\; \backslash mathbf\{v\}\; -\; \backslash mathbf\{v\}\_\backslash perp$$ where "$\backslash cdot$" denotes the dot product. Another formula for the tangential component is $$\backslash mathbf\{v\}\_\backslash parallel\; =\; -\backslash hat\backslash mathbf\{n\}\; \backslash times\; (\backslash hat\backslash mathbf\{n\}\backslash times\backslash mathbf\{v\}),$$

where "$\backslash times$" denotes the cross product.

These formulas do not depend on the particular unit normal $\backslash hat\backslash mathbf\{n\}$ 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 More generally, given a submanifold N of a manifold M and a point $p\; \backslash in\; N$, we get a short exact sequence involving the : $$T\_p\; N\; \backslash to\; T\_p\; M\; \backslash to\; T\_p\; M\; /\; T\_p\; N$$ The quotient space $T\_p\; M\; /\; T\_p\; N$ is a generalized space of normal vectors.

If M is a Riemannian manifold, the above sequence splits, and the tangent space of M at p decomposes as a direct sum of the component tangent to N and the component normal to N: $$T\_p\; M\; =\; T\_p\; N\; \backslash oplus\; N\_p\; N\; :=\; (T\_p\; N)^\backslash perp$$ Thus every tangent vector $v\; \backslash in\; T\_p\; M$ splits as where $v\_\backslash parallel\; \backslash in\; T\_p\; N$ and $v\_\backslash perp\; \backslash in\; N\_p\; N\; :=\; (T\_p\; N)^\backslash perp$.

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Computations Suppose N is given by non-degenerate equations.

If N is given explicitly, via parametric equations (such as a parametric curve), then the derivative gives a spanning set for the tangent bundle (it is a basis if and only if the parametrization is an immersion).

If N is given implicit surface (as in the above description of a surface, (or more generally as) a hypersurface) as a level set or intersection of level surface for $g\_i$, then the gradients of $g\_i$ span the normal space.

In both cases, we can again compute using the dot product; the cross product is special to 3 dimensions however.

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Applications

• Lagrange multipliers: constrained critical points are where the tangential component of the total derivative vanish.
• Surface normal
• Frenet–Serret formulas

• Vladimir Rojansky (1979). 9780486638348, New York: Dover Publications. ISBN 9780486638348

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