Sales rate

The Hopf circulation theorem , named after Heinz Hopf , is a mathematical theorem from elementary differential geometry . It is a clear and immediately understandable statement that a flat curve without self-penetrations can go around a point at most once.

Definitions

To define the polar angle function

Tangent directions

Let it be a continuously differentiable curve. Such a curve is called ${\displaystyle \gamma :[a,b]\rightarrow \mathbb {R} ^{2}\setminus \{0\}}$

• closed if, ${\displaystyle \gamma (a)=\gamma (b)}$
• regular , if the derivative is always not equal to 0,
• simple if is injective . ${\displaystyle \gamma |_{[a,b)}}$

If the Euclidean norm denotes the distance of a vector from 0, there is a continuous function so that ${\displaystyle \|\cdot \|}$ ${\displaystyle \varphi :[a,b]\rightarrow \mathbb {R} }$

${\displaystyle \gamma (t)=\|\gamma (t)\|\cdot (\cos \varphi (t),\sin \varphi (t))}$

for everyone . is the angle that the x-axis makes with the vector to the curve point . The specialty is that you can choose the so-called polar angle function continuously, that is, that the angle does not make any jumps. Then there is a quantity that measures how often the angle passes through a full angle . One can show that in the case of closed curves it is actually an integer that is called the revolution number of . ${\displaystyle t\in [a,b]}$ ${\displaystyle \varphi (t)}$ ${\displaystyle \gamma (t)}$ ${\displaystyle \varphi }$ ${\displaystyle \textstyle {\frac {1}{2\pi }}(\varphi (b)-\varphi (a))}$ ${\displaystyle W_{\gamma }}$ ${\displaystyle \gamma }$

Is now a twice continuously differentiable curve with tangent vector , so measures the circulation rate , how much full angle of the tangents turn at a curve circulation is called this figure the tangent revolution number of the curve , or their tangent speed. ${\displaystyle c:[a,b]\rightarrow \mathbb {R} ^{2}\setminus \{0\}}$ ${\displaystyle {\dot {c}}}$ ${\displaystyle W_{\dot {c}}}$ ${\displaystyle U_{c}}$ ${\displaystyle c}$

Formulation of the sentence

If there is a twice continuously differentiable, simply closed, regular curve, then applies . ${\displaystyle c:[a,b]\rightarrow \mathbb {R} ^{2}\setminus \{0\}}$ ${\displaystyle U_{c}=\pm 1}$

Remarks

A curve with a number of revolutions> 1 must have a colon.

Since the rotation number changes sign when the direction of rotation of the curve changes, the tangent rotation number in the above sentence cannot be further restricted.

If a continuously differentiable, closed, regular curve revolves around the zero point several times, that is, is , the curve cannot be simple according to the circulation theorem, that is, there must be at least one colon . This statement is vividly clear, because in order to reach the starting point again, you have to cross the parts of the curve that have already been circumnavigated. The mathematical proof essentially uses the existence of the polar angle function. ${\displaystyle |U_{c}|>1}$

As early as 1857 Bernhard Riemann wrote : “In the case of a simply connected surface spread over a finite part of the z-plane between the number of its simple branching points and the number of revolutions which the direction of its boundary line makes, the relation takes place that the latter around one Unit is greater than the former .; ... “With area we mean what is called a Riemann area today . A surface that lies entirely in the plane has no branch points, so the tangent speed (= number of revolutions made by the direction of its boundary line ) must be equal to 1. The theorem was first formulated and proven in the precise differential geometric form in 1916 by George Neville Watson . Today's standard proof goes back to Heinz Hopf, who also introduced the term circulation rate .

Individual evidence

1. ↑ Wolfgang Kühnel: Curves - Surfaces - Manifolds. Friedr. Vieweg & Sohn Verlag, 2008, ISBN 978-3-8348-0411-2 , sentence and definition 2.24
2. ↑ Wolfgang Kühnel: Curves - Surfaces - Manifolds. Friedr. Vieweg & Sohn Verlag, 2008, ISBN 978-3-8348-0411-2 , sentence 2.28
3. ↑ Christian Bär: Elementary Differential Geometry. Cambridge University Press 2010, ISBN 978-0-521-89671-9 , theorem 2.2.10
4. ↑ Bernhard Riemann: Theory of Abelian Functions. In: Journal for pure and applied mathematics. (1857), Vol. 54, pp. 101-155.
5. ↑ Heinz Hopf: About the rotation of the tangents and tendons of flat curves. Composito Math. (1935), Vol. 2, pp. 50-62.
6. ^ Peter Dombrowski: Differential Geometry. In: A Century of Mathematics 1890–1990: Festschrift for the anniversary of the DMV. P. 342.