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The unit circle is locally a $C^{1}$ graph at every point

Example

At every point $(a, b)$ of the unit circle $a^{2} + b^{2} = 1$ , the circle is locally the graph of a $C^{1}$ function of one coordinate.

Facts & Assumptions

Given: No hypotheses beyond those quantified in the statement.

[L1]

On an open domain, a $C^{1}$ map satisfying $F (a, b) = 0$ can be solved locally for the second block when $D_{y} F (a, b)$ is invertible (The Euclidean implicit function theorem with derivative formula).

[L2]

Direct difference quotients give $\partial_{x} F (x, y) = 2 x$ and $\partial_{y} F (x, y) = 2 y$ ; their displayed affine formulas are continuous. Thus the continuous-partials theorem gives $D F (x, y) (u, v) = 2 x u + 2 y v$ , and $F$ is $C^{1}$ (If all partial derivatives exist on a neighbourhood and are continuous at a point, then the map is totally differentiable there with Jacobian derivative, Continuously differentiable maps, local inverses, and local diffeomorphisms, For a natural $n \geq 1$ the function $x \mapsto x^{n}$ is differentiable everywhere with derivative $ι (n) x^{n - 1}$ ; for $n = 0$ it is the constant $1$ , with derivative $0$ ; for a natural $n \geq 1$ the function $x \mapsto x^{- n}$ is differentiable at every $x \neq 0$ with derivative $- ι (n) x^{- n - 1}$ ; consequently every polynomial function is differentiable at every real, with the derivative computed term by term).

Proof

technique · direct

1.1

Let $F (x, y) : = x^{2} + y^{2} - 1$ . Then [L2] gives $D F (x, y) (u, v) = 2 x u + 2 y v$ . At a point $(a, b)$ on the circle, at least one of $a, b$ is nonzero.

L2givenalgebra

2.1

If $b \neq 0$ , then $D_{y} F (a, b)$ is multiplication by $2 b$ and is invertible, so [L1] expresses the circle locally as $y = φ (x)$ . If $a \neq 0$ , exchange the coordinate blocks and obtain $x = ψ (y)$ . These alternatives cover every circle point by step 1.1.

step 1.1L1cases∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

29 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. Lebl, Basic Analysis II, §8.5.1 and Exercise 8.5.1 (standard reference, not scraped)

The unit circle is locally a C1C1 graph at every point