Leibniz sector formula

Curve with driving beam

closed curve with driving beam

The sector of formula Leibniz , named after Leibniz , calculates the area-oriented, the beam passes over a running curve of a portion, in particular it is possible with their surface areas of areas that are described by a closed curve, calculate.

formula

Be with a smooth curve , then it passes over the origin educated driving beam the oriented area of the following size: ${\displaystyle \gamma :[a,b]\rightarrow \mathbb {R} ^{2}}$ ${\displaystyle t\mapsto (x(t),y(t))}$ ${\displaystyle F}$

${\displaystyle F(\gamma )={\frac {1}{2}}\int _{a}^{b}(x(t)y^{\prime }(t)-y(t)x^{\prime }(t))dt}$

Slightly smooth curves

If there is a piecewise smooth curve on and a partition of , so that on the subintervals for is smooth, then: ${\displaystyle \gamma }$ ${\displaystyle [a,b]}$ ${\displaystyle \{t_{0},\ldots ,t_{n}\}}$ ${\displaystyle [a,b]}$ ${\displaystyle \gamma }$ ${\displaystyle [t_{k-1},t_{k}]}$ ${\displaystyle k=1,\ldots n}$

${\displaystyle F(\gamma )=F(\gamma _{1})+\ldots +F(\gamma _{n})}$

Here the curve restricted to the interval denotes . ${\displaystyle \gamma _{k}}$ ${\displaystyle [t_{k-1},t_{k}]}$

Connection with triangles

Triangle as a piecewise smooth curve

The sector formula can be understood as a generalization of the determinant formula for calculating the area of ​​triangles. Are , , the vertices of any triangle, then this is expressed by the following piecewise smooth curve described: ${\displaystyle A=(x_{1},y_{1})}$ ${\displaystyle B=(x_{2},y_{2})}$ ${\displaystyle C=(x_{3},y_{3})}$ ${\displaystyle [0,3]\rightarrow \mathbb {R} ^{2}}$

${\displaystyle \gamma (t):={\begin{cases}\gamma _{a}(t)=(x_{1}+(x_{2}-x_{1})t,\,y_{1}+(y_{2}-y_{1})t)&{\text{falls }}0\leq t\leq 1\\\gamma _{b}(t)=(x_{2}+(x_{3}-x_{2})(t-1),\,y_{2}+(y_{3}-y_{2})(t-1))&{\text{falls }}1\leq t\leq 2\\\gamma _{c}(t)=(x_{3}+(x_{1}-x_{3})(t-2),\,y_{3}+(y_{1}-y_{3})(t-2))&{\text{falls }}2\leq t\leq 3\end{cases}}}$

Then the following applies to the area calculation of the triangle:

${\displaystyle {\begin{aligned}F(\triangle )&={\frac {1}{2}}\left|{\begin{matrix}1&x_{1}&y_{1}\\1&x_{2}&y_{2}\\1&x_{3}&y_{3}\end{matrix}}\right|={\frac {1}{2}}[(x_{1}y_{2}-y_{1}x_{2})+(x_{2}y_{3}-y_{2}x_{3})+(x_{3}y_{1}-y_{3}x_{1})]\\&=\quad {\frac {1}{2}}\int _{0}^{1}[(x_{1}+(x_{2}-x_{1})t)(y_{2}-y_{1})-(y_{1}+(y_{2}-y_{1})t)(x_{2}-x_{1})]dt\\&\quad +{\frac {1}{2}}\int _{1}^{2}[(x_{2}+(x_{3}-x_{2})(t-1))(y_{3}-y_{2})-(y_{2}+(y_{3}-y_{2})(t-1))(x_{3}-x_{2})]dt\\&\quad +{\frac {1}{2}}\int _{2}^{3}[(x_{3}+(x_{1}-x_{3})(t-2))(y_{1}-y_{3})-(y_{3}+(y_{1}-y_{3})(t-2))(x_{1}-x_{3})]dt\\&=F(\gamma _{a})+F(\gamma _{b})+F(\gamma _{c})\\&=F(\gamma )\end{aligned}}}$

Connection with the integral sentences

In the case of a closed curve, Leibniz's sector formula results as a special case of Green's integral theorem . The integral theorem for a curve with enclosed area and two differentiable functions the following equation: ${\displaystyle \gamma :[a,b]\rightarrow \mathbb {R} ^{2}}$ ${\displaystyle \gamma (a)=\gamma (b)}$ ${\displaystyle B}$ ${\displaystyle f,g:\mathbb {R} ^{2}\rightarrow \mathbb {R} }$

${\displaystyle \int _{B}{\Bigl [}g_{x}(x,y)-f_{y}(x,y){\Bigr ]}dxdy=\int _{a}^{b}{\Bigl [}f{\bigl (}x(t),y(t){\bigr )}\cdot x^{\prime }(t)+g{\bigl (}x(t),y(t){\bigr )}\cdot y^{\prime }(t){\Bigr ]}dt}$

If we choose for the local functions and so true and and you get: ${\displaystyle f(x,y)=-y}$ ${\displaystyle g(x,y)=x}$ ${\displaystyle f_{y}(x,y)=-1}$ ${\displaystyle g_{x}(x,y)=1}$

${\displaystyle {\begin{aligned}&\int _{B}(1-(-1))dxdy=\int _{a}^{b}{\Bigl [}-y(t)\cdot x^{\prime }(t)+x(t)\cdot y^{\prime }(t){\Bigr ]}dt\\\Leftrightarrow &\int _{B}1dxdy={\frac {1}{2}}\int _{a}^{b}{\Bigl [}x(t)\cdot y^{\prime }(t)-y(t)\cdot x^{\prime }(t){\Bigr ]}dt\end{aligned}}}$

Since the integration over an area with 1 provides the area itself, the following applies:

${\displaystyle F(\gamma )=\int _{B}1dxdy={\frac {1}{2}}\int _{a}^{b}{\Bigl [}x(t)\cdot y^{\prime }(t)-y(t)\cdot x^{\prime }(t){\Bigr ]}dt}$.

Alternative formula

Alternative formula

In the literature, another formula is occasionally referred to as Leibniz's sector formula. This is much more special and instead of coordinate functions and the parameter curve uses a function that describes the distance between a curve point and the center of a star-shaped set . With this then applies: ${\displaystyle x(t)}$ ${\displaystyle y(t)}$ ${\displaystyle \gamma (t)}$ ${\displaystyle r(t)}$ ${\displaystyle B}$

${\displaystyle F(B)={\frac {1}{2}}\int _{a}^{b}r(t)^{2}\ dt}$

Since this formula does not use an oriented area in contrast to the previous one, it is only valid for star-shaped sets. If there is a center of the star-shaped set, then r (t) can be calculated using the relationship from the coordinate functions of the parameter curve. ${\displaystyle (x_{z},y_{z})}$ ${\displaystyle r(t)={\sqrt {(x(t)-x_{z})^{2}+(y(t)-y_{z})^{2}}}}$

example

A heart curve has the following parameter representation: ${\displaystyle \gamma :[0,2\pi ]\rightarrow \mathbb {R} ^{2}}$

${\displaystyle x=a\cos(t)(1+\cos(t))\qquad y=a\sin(t)(1+\cos(t))}$

The sector formula then gives the following area:

${\displaystyle {\begin{aligned}F(\gamma )&={\frac {1}{2}}\int _{0}^{2\pi }{\Bigl [}a\cos(t)(1+\cos(t))a(\cos(t)+2\cos(t)^{2}-1)+a\sin(t)(1+2\cos(t))a\sin(t)(1+\cos(t)){\Bigr ]}dt\\&={\frac {1}{2}}\int _{0}^{2\pi }{\Bigl [}(1+\cos(t))^{2}a^{2}{\Bigr ]}dt\\&={\frac {3}{2}}a^{2}\pi \end{aligned}}}$

Heart curve

When using the alternative formula one can choose as the center and then get: ${\displaystyle (0,0)}$

${\displaystyle {\begin{aligned}F(\gamma )&={\frac {1}{2}}\int _{0}^{2\pi }{\Bigl [}{\bigl (}a\cos(t)(1+\cos(t)){\bigr )}^{2}+{\bigl (}a\sin(t)(1+\cos(t)){\bigr )}^{2}{\Bigr ]}dt\\&={\frac {1}{2}}\int _{0}^{2\pi }{\Bigl [}(1+\cos(t))^{2}a^{2}(\cos(t)^{2}+\sin(t)^{2}){\Bigr ]}dt\\&={\frac {1}{2}}\int _{0}^{2\pi }{\Bigl [}(1+\cos(t))^{2}a^{2}{\Bigr ]}dt\\&={\frac {3}{2}}a^{2}\pi \end{aligned}}}$

literature

• Konrad Köngisberger: Analysis 1 . 2nd edition, Springer 1992, ISBN 3-540-55116-6 , p. 343
• Wolfgang Walter: Analysis I . 2nd edition, Springer 1985, ISBN 3-540-51708-1 , pp. 285-286
• Harro Heuser: Textbook of Analysis - Part 2 . 5th edition, Teubner 1990, ISBN 3-519-42222-0 , p. 498

Web links

• J. Weikert: Curves and arc length (PDF; 595 kB). In mathematics for computer scientists I , script Uni Saarland
• Multi-dimensional integral calculus . In higher mathematics II (university script)