# Length

A numerical characteristic of the extension of lines in a metric space. The length of a segment of a straight line is the distance between its ends, measured by means of some segment accepted as the unit length. The length of a broken line is the sum of the lengths of its parts. The length of a simple arc is the least upper bound of the lengths of the broken lines inscribed in this arc. Any continuous curve has a length, finite or infinite. If its length is finite, the curve is known as rectifiable. The length of a planar curve defined in rectangular coordinates by an equation $y = f( x)$, $a \leq x \leq b$( $f$ having a continuous derivative $f ^ { \prime }$) is given by the integral

$$s = \int\limits _ { a } ^ { b } \sqrt {1 + [ f ^ { \prime } ( x) ] ^ {2} } d x .$$

If the curve is given in parametric form

$$x = x ( t) ,\ y = y ( t) ,\ t _ {1} \leq t \leq t _ {2} ,$$

its length is given by

$$s = \int\limits _ { t _ {1} } ^ { {t _ 2 } } \sqrt {[ x ^ \prime ( t) ] ^ {2} + [ y ^ \prime ( t) ] ^ {2} } d t .$$

The length of a rectifiable curve does not depend on the parametrization. The length of a spatial curve given in parametric form $x = x ( t)$, $y = y ( t)$, $z = z ( t)$, $t _ {1} \leq t \leq t _ {2}$, is given by the formula

$$s = \int\limits _ { t _ {1} } ^ { {t _ 2 } } \sqrt {[ x ^ \prime ( t) ] ^ {2} + [ y ^ \prime ( t) ] ^ {2} + [ z ^ \prime ( t) ] ^ {2} } d t .$$

In the case of an $n$- dimensional space,

$$s = \int\limits _ { t _ {1} } ^ { {t _ 2 } } \sqrt {\sum _ {i = 1 } ^ { n } [ x _ {i} ^ \prime ( t) ] ^ {2} } d t .$$

Let $\gamma$ be a continuously-differentiable curve, given by functions $u = u ( t)$, $v= v ( t)$, on a continuously-differentiable surface $\mathbf r = \mathbf r ( u , v )$. Then the length of an arc of the curve counted from the point corresponding to the parameter value $t = t _ {0}$ is equal to

$$s ( t , t _ {0} ) = \int\limits _ {t _ {0} } ^ { t } | \mathbf r ^ \prime ( t) | dt = \int\limits _ {\gamma ( P _ {0} , P ) } | d \mathbf r ( u , v ) | = \int\limits _ {t _ {0} } ^ { t } \sqrt I ,$$

where $I$ is the first fundamental form of the surface. The length of a continuously-differentiable curve given by functions $x ^ {i} = x ^ {i} ( t)$, $t _ {1} \leq t \leq t _ {0}$, in a Riemannian space with metric tensor $g _ {ik}$ is

$$s = \int\limits _ { t _ {1} } ^ { {t _ 2 } } d s = \int\limits _ { t _ {1} } ^ { {t _ 2 } } \sqrt {g _ {ik} \frac{d x ^ {i} }{dt} \frac{d x ^ {k} }{dt} } d t .$$