Alphabeta MathHow statement and proof provenance work
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The tangent space to a regular level set
Definition
Let be , let be a regular value, and let (Regular and critical points, regular and critical values, and level sets). For regular, . This kernel (Kernel and image of a linear map) is the tangent space to the regular level set at .
Since is surjective, rank-nullity gives (Rank-nullity: ). Thus the definition introduces an existing linear subspace of the asserted dimension; it makes no assignment when the fibre is empty because there is then no point .
Depends on
Used by
- A Euclidean sphere is a regular level set with tangent hyperplanes Example
- A positive-definite quadratic ellipsoid is a regular level set Example
- The graph of a Cᵏ Euclidean map is a regular level set Example
- The one-sheeted hyperboloid is a regular surface of revolution Example
- The orthogonal group is a regular level set of dimension n(n-1)/2 Example
- Regular level surfaces have local regular parametrizations with the same tangent plane Theorem
- Tangent vectors to a regular level set are exactly its curve velocities Theorem
Dependency tree · two levels
10 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- J. M. Lee, Introduction to Smooth Manifolds, Theorem 8.8 (standard reference, not scraped)
- L. W. Tu, An Introduction to Manifolds, Section 11.2 (standard reference, not scraped)