Misplaced Pages

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71-399: When the dam was built, the decision was made to use the dam to store and regulate water only and not to provide energy. Construction started on the dam in 1968 and finished in 1972. The dam is 134 m (440 ft) high and 52.3 m (171.6 ft) wide at the foundation. The reservoir capacity is 424,000,000 m (344,000 acre feet). It is a double curvature concrete arch buttress design. Monitoring of

142-407: A b | f ′ ( t ) |   d t . {\displaystyle L(f)=\lim _{N\to \infty }\sum _{i=1}^{N}{\bigg |}f(t_{i})-f(t_{i-1}){\bigg |}=\lim _{N\to \infty }\sum _{i=1}^{N}\left|{\frac {f(t_{i})-f(t_{i-1})}{\Delta t}}\right|\Delta t=\int _{a}^{b}{\Big |}f'(t){\Big |}\ dt.} The last equality is proved by the following steps: With

213-899: A b | g ′ ( φ ( t ) ) φ ′ ( t ) |   d t = ∫ a b | g ′ ( φ ( t ) ) | φ ′ ( t )   d t in the case  φ  is non-decreasing = ∫ c d | g ′ ( u ) |   d u using integration by substitution = L ( g ) . {\displaystyle {\begin{aligned}L(f)&=\int _{a}^{b}{\Big |}f'(t){\Big |}\ dt=\int _{a}^{b}{\Big |}g'(\varphi (t))\varphi '(t){\Big |}\ dt\\&=\int _{a}^{b}{\Big |}g'(\varphi (t)){\Big |}\varphi '(t)\ dt\quad {\text{in

284-564: A ) {\displaystyle \varepsilon N\Delta t=\varepsilon (b-a)} , and N > ( b − a ) / δ ( ε ) {\displaystyle N>(b-a)/\delta (\varepsilon )} . In the limit N → ∞ , {\displaystyle N\to \infty ,} δ ( ε ) → 0 {\displaystyle \delta (\varepsilon )\to 0} so ε → 0 {\displaystyle \varepsilon \to 0} thus

355-1301: A ) / δ ( ε ) {\textstyle N>(b-a)/\delta (\varepsilon )} so that Δ t < δ ( ε ) {\displaystyle \Delta t<\delta (\varepsilon )} , it becomes Δ t ∑ i = 1 N ( | ∫ 0 1 f ′ ( t i − 1 + θ ( t i − t i − 1 ) )   d θ | − | f ′ ( t i ) | ) < ε N Δ t {\displaystyle \Delta t\sum _{i=1}^{N}\left(\left|\int _{0}^{1}f'(t_{i-1}+\theta (t_{i}-t_{i-1}))\ d\theta \right|-\left|f'(t_{i})\right|\right)<\varepsilon N\Delta t} with | f ′ ( t i ) | = ∫ 0 1 | f ′ ( t i ) | d θ {\displaystyle \left|f'(t_{i})\right|=\int _{0}^{1}\left|f'(t_{i})\right|d\theta } , ε N Δ t = ε ( b −

426-509: A + i ( b − a ) / N = a + i Δ t {\displaystyle t_{i}=a+i(b-a)/N=a+i\Delta t} with Δ t = b − a N = t i − t i − 1 {\displaystyle \Delta t={\frac {b-a}{N}}=t_{i}-t_{i-1}} for i = 0 , 1 , … , N . {\displaystyle i=0,1,\dotsc ,N.} This definition

497-661: A , b ] → R n {\displaystyle f:[a,b]\to \mathbb {R} ^{n}} on [ a , b ] {\displaystyle [a,b]} is always finite, i.e., rectifiable . The definition of arc length of a smooth curve as the integral of the norm of the derivative is equivalent to the definition L ( f ) = sup ∑ i = 1 N | f ( t i ) − f ( t i − 1 ) | {\displaystyle L(f)=\sup \sum _{i=1}^{N}{\bigg |}f(t_{i})-f(t_{i-1}){\bigg |}} where

568-407: A closed form solution for the arc length of an elliptic and hyperbolic arc led to the development of the elliptic integrals . In most cases, including even simple curves, there are no closed-form solutions for arc length and numerical integration is necessary. Numerical integration of the arc length integral is usually very efficient. For example, consider the problem of finding the length of

639-432: A coordinate-free way as These formulas can be derived from the special case of arc-length parametrization in the following way. The above condition on the parametrisation imply that the arc length s is a differentiable monotonic function of the parameter t , and conversely that t is a monotonic function of s . Moreover, by changing, if needed, s to – s , one may suppose that these functions are increasing and have

710-817: A curve expressed in polar coordinates, the arc length is: ∫ t 1 t 2 ( d r d t ) 2 + r 2 ( d θ d t ) 2 d t = ∫ θ ( t 1 ) θ ( t 2 ) ( d r d θ ) 2 + r 2 d θ . {\displaystyle \int _{t_{1}}^{t_{2}}{\sqrt {\left({\frac {dr}{dt}}\right)^{2}+r^{2}\left({\frac {d\theta }{dt}}\right)^{2}\,}}dt=\int _{\theta (t_{1})}^{\theta (t_{2})}{\sqrt {\left({\frac {dr}{d\theta }}\right)^{2}+r^{2}\,}}d\theta .} The second expression

781-403: A curve expressed in polar coordinates. The mapping that transforms from polar coordinates to rectangular coordinates is x ( r , θ ) = ( r cos ⁡ θ , r sin ⁡ θ ) . {\displaystyle \mathbf {x} (r,\theta )=(r\cos \theta ,r\sin \theta ).} The integrand of the arc length integral

SECTION 10

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852-788: A curve expressed in spherical coordinates where θ {\displaystyle \theta } is the polar angle measured from the positive z {\displaystyle z} -axis and ϕ {\displaystyle \phi } is the azimuthal angle. The mapping that transforms from spherical coordinates to rectangular coordinates is x ( r , θ , ϕ ) = ( r sin ⁡ θ cos ⁡ ϕ , r sin ⁡ θ sin ⁡ ϕ , r cos ⁡ θ ) . {\displaystyle \mathbf {x} (r,\theta ,\phi )=(r\sin \theta \cos \phi ,r\sin \theta \sin \phi ,r\cos \theta ).} Using

923-696: A curve expressed in spherical coordinates, the arc length is ∫ t 1 t 2 ( d r d t ) 2 + r 2 ( d θ d t ) 2 + r 2 sin 2 ⁡ θ ( d ϕ d t ) 2 d t . {\displaystyle \int _{t_{1}}^{t_{2}}{\sqrt {\left({\frac {dr}{dt}}\right)^{2}+r^{2}\left({\frac {d\theta }{dt}}\right)^{2}+r^{2}\sin ^{2}\theta \left({\frac {d\phi }{dt}}\right)^{2}\,}}dt.} A very similar calculation shows that

994-517: A general formula that provides closed-form solutions in some cases. A curve in the plane can be approximated by connecting a finite number of points on the curve using (straight) line segments to create a polygonal path . Since it is straightforward to calculate the length of each linear segment (using the Pythagorean theorem in Euclidean space, for example), the total length of

1065-411: A positive derivative. Using notation of the preceding section and the chain rule , one has and thus, by taking the norm of both sides where the prime denotes differentiation with respect to t . The curvature is the norm of the derivative of T with respect to s . By using the above formula and the chain rule this derivative and its norm can be expressed in terms of γ ′ and γ ″ only, with

1136-434: A quarter of the circle. Since d y / d x = − x / 1 − x 2 {\textstyle dy/dx=-x{\big /}{\sqrt {1-x^{2}}}} and 1 + ( d y / d x ) 2 = 1 / ( 1 − x 2 ) , {\displaystyle 1+(dy/dx)^{2}=1{\big /}\left(1-x^{2}\right),}

1207-481: A quarter of the unit circle by numerically integrating the arc length integral. The upper half of the unit circle can be parameterized as y = 1 − x 2 . {\displaystyle y={\sqrt {1-x^{2}}}.} The interval x ∈ [ − 2 / 2 , 2 / 2 ] {\displaystyle x\in \left[-{\sqrt {2}}/2,{\sqrt {2}}/2\right]} corresponds to

1278-401: A twice differentiable plane curve. Here proper means that on the domain of definition of the parametrization, the derivative ⁠ d γ / dt ⁠ is defined, differentiable and nowhere equal to the zero vector. With such a parametrization, the signed curvature is where primes refer to derivatives with respect to t . The curvature κ is thus These can be expressed in

1349-711: Is ( x u u ′ + x v v ′ ) ⋅ ( x u u ′ + x v v ′ ) = g 11 ( u ′ ) 2 + 2 g 12 u ′ v ′ + g 22 ( v ′ ) 2 {\displaystyle \left(\mathbf {x} _{u}u'+\mathbf {x} _{v}v'\right)\cdot (\mathbf {x} _{u}u'+\mathbf {x} _{v}v')=g_{11}\left(u'\right)^{2}+2g_{12}u'v'+g_{22}\left(v'\right)^{2}} (where g i j {\displaystyle g_{ij}}

1420-543: Is | ( x ∘ C ) ′ ( t ) | . {\displaystyle \left|\left(\mathbf {x} \circ \mathbf {C} \right)'(t)\right|.} The chain rule for vector fields shows that D ( x ∘ C ) = x r r ′ + x θ θ ′ . {\displaystyle D(\mathbf {x} \circ \mathbf {C} )=\mathbf {x} _{r}r'+\mathbf {x} _{\theta }\theta '.} So

1491-620: Is continuously differentiable , then it is simply a special case of a parametric equation where x = t {\displaystyle x=t} and y = f ( t ) . {\displaystyle y=f(t).} The Euclidean distance of each infinitesimal segment of the arc can be given by: d x 2 + d y 2 = 1 + ( d y d x ) 2 d x . {\displaystyle {\sqrt {dx^{2}+dy^{2}}}={\sqrt {1+\left({\frac {dy}{dx}}\right)^{2}\,}}dx.} The arc length

SECTION 20

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1562-474: Is (assuming 𝜿 ( s ) ≠ 0) and the center of curvature is on the normal to the curve, the center of curvature is the point (In case the curvature is zero, the center of curvature is not located anywhere on the plane R and is often said to be located "at infinity".) If N ( s ) is the unit normal vector obtained from T ( s ) by a counterclockwise rotation of ⁠ π / 2 ⁠ , then with k ( s ) = ± κ ( s ) . The real number k ( s )

1633-445: Is 500 rpm. The hydraulic head is 56 m. Maximum flow per turbine is 8 m³/s. Curvature In mathematics , curvature is any of several strongly related concepts in geometry that intuitively measure the amount by which a curve deviates from being a straight line or by which a surface deviates from being a plane . If a curve or surface is contained in a larger space, curvature can be defined extrinsically relative to

1704-430: Is another continuously differentiable parameterization of the curve originally defined by f . {\displaystyle f.} The arc length of the curve is the same regardless of the parameterization used to define the curve: L ( f ) = ∫ a b | f ′ ( t ) |   d t = ∫

1775-471: Is called rectification of a curve. The lengths of the successive approximations will not decrease and may keep increasing indefinitely, but for smooth curves they will tend to a finite limit as the lengths of the segments get arbitrarily small . For some curves, there is a smallest number L {\displaystyle L} that is an upper bound on the length of all polygonal approximations (rectification). These curves are called rectifiable and

1846-418: Is called the oriented curvature or signed curvature . It depends on both the orientation of the plane (definition of counterclockwise), and the orientation of the curve provided by the parametrization. In fact, the change of variable s → – s provides another arc-length parametrization, and changes the sign of k ( s ) . Let γ ( t ) = ( x ( t ), y ( t )) be a proper parametric representation of

1917-414: Is difficult to manipulate and to express in formulas. Therefore, other equivalent definitions have been introduced. Every differentiable curve can be parametrized with respect to arc length . In the case of a plane curve, this means the existence of a parametrization γ ( s ) = ( x ( s ), y ( s )) , where x and y are real-valued differentiable functions whose derivatives satisfy This means that

1988-597: Is equivalent to the standard definition of arc length as an integral: L ( f ) = lim N → ∞ ∑ i = 1 N | f ( t i ) − f ( t i − 1 ) | = lim N → ∞ ∑ i = 1 N | f ( t i ) − f ( t i − 1 ) Δ t | Δ t = ∫

2059-420: Is for a polar graph r = r ( θ ) {\displaystyle r=r(\theta )} parameterized by t = θ {\displaystyle t=\theta } . Now let C ( t ) = ( r ( t ) , θ ( t ) , ϕ ( t ) ) {\displaystyle \mathbf {C} (t)=(r(t),\theta (t),\phi (t))} be

2130-409: Is more complex, as it depends on the choice of a direction on the surface or manifold. This leads to the concepts of maximal curvature , minimal curvature , and mean curvature . In Tractatus de configurationibus qualitatum et motuum, the 14th-century philosopher and mathematician Nicole Oresme introduces the concept of curvature as a measure of departure from straightness; for circles he has

2201-411: Is often given as a definition of the curvature. Historically, the curvature of a differentiable curve was defined through the osculating circle , which is the circle that best approximates the curve at a point. More precisely, given a point P on a curve, every other point Q of the curve defines a circle (or sometimes a line) passing through Q and tangent to the curve at P . The osculating circle

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2272-426: Is possible to evaluate this integral to almost machine precision with only 16 integrand evaluations. Let x ( u , v ) {\displaystyle \mathbf {x} (u,v)} be a surface mapping and let C ( t ) = ( u ( t ) , v ( t ) ) {\displaystyle \mathbf {C} (t)=(u(t),v(t))} be a curve on this surface. The integrand of

2343-399: Is somewhat arbitrary, as it depends on the orientation of the curve. In the case of the graph of a function, there is a natural orientation by increasing values of x . This makes significant the sign of the signed curvature. The sign of the signed curvature is the same as the sign of the second derivative of f . If it is positive then the graph has an upward concavity, and, if it is negative

2414-669: Is the first fundamental form coefficient), so the integrand of the arc length integral can be written as g a b ( u a ) ′ ( u b ) ′ {\displaystyle {\sqrt {g_{ab}\left(u^{a}\right)'\left(u^{b}\right)'\,}}} (where u 1 = u {\displaystyle u^{1}=u} and u 2 = v {\displaystyle u^{2}=v} ). Let C ( t ) = ( r ( t ) , θ ( t ) ) {\displaystyle \mathbf {C} (t)=(r(t),\theta (t))} be

2485-426: Is the limit , if it exists, of this circle when Q tends to P . Then the center and the radius of curvature of the curve at P are the center and the radius of the osculating circle. The curvature is the reciprocal of radius of curvature. That is, the curvature is where R is the radius of curvature (the whole circle has this curvature, it can be read as turn 2π over the length 2π R ). This definition

2556-455: Is then given by: s = ∫ a b 1 + ( d y d x ) 2 d x . {\displaystyle s=\int _{a}^{b}{\sqrt {1+\left({\frac {dy}{dx}}\right)^{2}\,}}dx.} Curves with closed-form solutions for arc length include the catenary , circle , cycloid , logarithmic spiral , parabola , semicubical parabola and straight line . The lack of

2627-517: The arc length is defined as the number L {\displaystyle L} . A signed arc length can be defined to convey a sense of orientation or "direction" with respect to a reference point taken as origin in the curve (see also: curve orientation and signed distance ). Let f : [ a , b ] → R n {\displaystyle f\colon [a,b]\to \mathbb {R} ^{n}} be an injective and continuously differentiable (i.e.,

2698-410: The supremum is taken over all possible partitions a = t 0 < t 1 < ⋯ < t N − 1 < t N = b {\displaystyle a=t_{0}<t_{1}<\dots <t_{N-1}<t_{N}=b} of [ a , b ] . {\displaystyle [a,b].} This definition as the supremum of

2769-444: The wave equation of a string under tension, and other applications where small slopes are involved. This often allows systems that are otherwise nonlinear to be treated approximately as linear. If a curve is defined in polar coordinates by the radius expressed as a function of the polar angle, that is r is a function of θ , then its curvature is where the prime refers to differentiation with respect to θ . This results from

2840-2618: The above step result, it becomes ∑ i = 1 N | ∫ 0 1 f ′ ( t i − 1 + θ ( t i − t i − 1 ) )   d θ | Δ t − ∑ i = 1 N | f ′ ( t i ) | Δ t . {\displaystyle \sum _{i=1}^{N}\left|\int _{0}^{1}f'(t_{i-1}+\theta (t_{i}-t_{i-1}))\ d\theta \right|\Delta t-\sum _{i=1}^{N}\left|f'(t_{i})\right|\Delta t.} Terms are rearranged so that it becomes Δ t ∑ i = 1 N ( | ∫ 0 1 f ′ ( t i − 1 + θ ( t i − t i − 1 ) )   d θ | − ∫ 0 1 | f ′ ( t i ) | d θ ) ≦ Δ t ∑ i = 1 N ( ∫ 0 1 | f ′ ( t i − 1 + θ ( t i − t i − 1 ) ) |   d θ − ∫ 0 1 | f ′ ( t i ) | d θ ) = Δ t ∑ i = 1 N ∫ 0 1 | f ′ ( t i − 1 + θ ( t i − t i − 1 ) ) | − | f ′ ( t i ) |   d θ {\displaystyle {\begin{aligned}&\Delta t\sum _{i=1}^{N}\left(\left|\int _{0}^{1}f'(t_{i-1}+\theta (t_{i}-t_{i-1}))\ d\theta \right|-\int _{0}^{1}\left|f'(t_{i})\right|d\theta \right)\\&\qquad \leqq \Delta t\sum _{i=1}^{N}\left(\int _{0}^{1}\left|f'(t_{i-1}+\theta (t_{i}-t_{i-1}))\right|\ d\theta -\int _{0}^{1}\left|f'(t_{i})\right|d\theta \right)\\&\qquad =\Delta t\sum _{i=1}^{N}\int _{0}^{1}\left|f'(t_{i-1}+\theta (t_{i}-t_{i-1}))\right|-\left|f'(t_{i})\right|\ d\theta \end{aligned}}} where in

2911-605: The all possible partition sums is also valid if f {\displaystyle f} is merely continuous, not differentiable. A curve can be parameterized in infinitely many ways. Let φ : [ a , b ] → [ c , d ] {\displaystyle \varphi :[a,b]\to [c,d]} be any continuously differentiable bijection . Then g = f ∘ φ − 1 : [ c , d ] → R n {\displaystyle g=f\circ \varphi ^{-1}:[c,d]\to \mathbb {R} ^{n}}

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2982-450: The ambient space. Curvature of Riemannian manifolds of dimension at least two can be defined intrinsically without reference to a larger space. For curves, the canonical example is that of a circle , which has a curvature equal to the reciprocal of its radius . Smaller circles bend more sharply, and hence have higher curvature. The curvature at a point of a differentiable curve is the curvature of its osculating circle — that is,

3053-431: The approximation can be found by summation of the lengths of each linear segment; that approximation is known as the (cumulative) chordal distance . If the curve is not already a polygonal path, then using a progressively larger number of line segments of smaller lengths will result in better curve length approximations. Such a curve length determination by approximating the curve as connected (straight) line segments

3124-792: The arc length integral is | ( x ∘ C ) ′ ( t ) | . {\displaystyle \left|\left(\mathbf {x} \circ \mathbf {C} \right)'(t)\right|.} Evaluating the derivative requires the chain rule for vector fields: D ( x ∘ C ) = ( x u   x v ) ( u ′ v ′ ) = x u u ′ + x v v ′ . {\displaystyle D(\mathbf {x} \circ \mathbf {C} )=(\mathbf {x} _{u}\ \mathbf {x} _{v}){\binom {u'}{v'}}=\mathbf {x} _{u}u'+\mathbf {x} _{v}v'.} The squared norm of this vector

3195-409: The arc-length parameter s completely eliminated, giving the above formulas for the curvature. The graph of a function y = f ( x ) , is a special case of a parametrized curve, of the form As the first and second derivatives of x are 1 and 0, previous formulas simplify to for the curvature, and to for the signed curvature. In the general case of a curve, the sign of the signed curvature

3266-432: The case }}\varphi {\text{ is non-decreasing}}\\&=\int _{c}^{d}{\Big |}g'(u){\Big |}\ du\quad {\text{using integration by substitution}}\\&=L(g).\end{aligned}}} If a planar curve in R 2 {\displaystyle \mathbb {R} ^{2}} is defined by the equation y = f ( x ) , {\displaystyle y=f(x),} where f {\displaystyle f}

3337-686: The chain rule again shows that D ( x ∘ C ) = x r r ′ + x θ θ ′ + x ϕ ϕ ′ . {\displaystyle D(\mathbf {x} \circ \mathbf {C} )=\mathbf {x} _{r}r'+\mathbf {x} _{\theta }\theta '+\mathbf {x} _{\phi }\phi '.} All dot products x i ⋅ x j {\displaystyle \mathbf {x} _{i}\cdot \mathbf {x} _{j}} where i {\displaystyle i} and j {\displaystyle j} differ are zero, so

3408-413: The circle that best approximates the curve near this point. The curvature of a straight line is zero. In contrast to the tangent , which is a vector quantity, the curvature at a point is typically a scalar quantity, that is, it is expressed by a single real number . For surfaces (and, more generally for higher-dimensional manifolds ), that are embedded in a Euclidean space , the concept of curvature

3479-425: The curvature as being inversely proportional to the radius; and he attempts to extend this idea to other curves as a continuously varying magnitude. The curvature of a differentiable curve was originally defined through osculating circles . In this setting, Augustin-Louis Cauchy showed that the center of curvature is the intersection point of two infinitely close normal lines to the curve. curve Intuitively,

3550-423: The curvature can be derived from the expression of the curvature of the graph of a function by using the implicit function theorem and the fact that, on such a curve, one has It can be useful to verify on simple examples that the different formulas given in the preceding sections give the same result. A common parametrization of a circle of radius r is γ ( t ) = ( r cos t , r sin t ) . The formula for

3621-451: The curvature describes for any part of a curve how much the curve direction changes over a small distance travelled (e.g. angle in rad/m ), so it is a measure of the instantaneous rate of change of direction of a point that moves on the curve: the larger the curvature, the larger this rate of change. In other words, the curvature measures how fast the unit tangent vector to the curve at point p rotates when point p moves at unit speed along

SECTION 50

#1732765264406

3692-441: The curvature gives It follows, as expected, that the radius of curvature is the radius of the circle, and that the center of curvature is the center of the circle. The circle is a rare case where the arc-length parametrization is easy to compute, as it is It is an arc-length parametrization, since the norm of is equal to one. This parametrization gives the same value for the curvature, as it amounts to division by r in both

3763-443: The curve at P ( s ) , which is also the derivative of P ( s ) with respect to s . Then, the derivative of T ( s ) with respect to s is a vector that is normal to the curve and whose length is the curvature. To be meaningful, the definition of the curvature and its different characterizations require that the curve is continuously differentiable near P , for having a tangent that varies continuously; it requires also that

3834-421: The curve defined by F ( x , y ) = 0 , but it would change the sign of the numerator if the absolute value were omitted in the preceding formula. A point of the curve where F x = F y = 0 is a singular point , which means that the curve is not differentiable at this point, and thus that the curvature is not defined (most often, the point is either a crossing point or a cusp ). The above formula for

3905-436: The curve has a finite length). If a curve can be parameterized as an injective and continuously differentiable function (i.e., the derivative is a continuous function) f : [ a , b ] → R n {\displaystyle f\colon [a,b]\to \mathbb {R} ^{n}} , then the curve is rectifiable (i.e., it has a finite length). The advent of infinitesimal calculus led to

3976-446: The curve is twice differentiable at P , for insuring the existence of the involved limits, and of the derivative of T ( s ) . The characterization of the curvature in terms of the derivative of the unit tangent vector is probably less intuitive than the definition in terms of the osculating circle, but formulas for computing the curvature are easier to deduce. Therefore, and also because of its use in kinematics , this characterization

4047-400: The curve. In fact, it can be proved that this instantaneous rate of change is exactly the curvature. More precisely, suppose that the point is moving on the curve at a constant speed of one unit, that is, the position of the point P ( s ) is a function of the parameter s , which may be thought as the time or as the arc length from a given origin. Let T ( s ) be a unit tangent vector of

4118-453: The dam had caused fracturing in the rock, resulting in significantly increasing the foundation's permeability . The crack has been treated and since then the problems have abated. The dam supports a hydroelectric power plant with a nameplate capacity of 9,56 MW . Its annual generation lies between 18,66 (2012) and 44,49 (1998) GWh . The power station contains 2 Francis turbine -generators with 4,78 MW (5,4 MVA ) each. The turbine rotation

4189-467: The dam revealed abnormal movement. Although dams normally move, the left side of the El Atazar Dam was moving more than the right because a support built on the dam's right made that side less flexible. In 1977 a crack was noticed in the dam. By 1979 the crack had grown to 46 m (150 ft) in length and was repaired. Inspection in 1983 revealed that the settling in the foundations and the movements of

4260-750: The derivative is a continuous function) function. The length of the curve defined by f {\displaystyle f} can be defined as the limit of the sum of linear segment lengths for a regular partition of [ a , b ] {\displaystyle [a,b]} as the number of segments approaches infinity. This means L ( f ) = lim N → ∞ ∑ i = 1 N | f ( t i ) − f ( t i − 1 ) | {\displaystyle L(f)=\lim _{N\to \infty }\sum _{i=1}^{N}{\bigg |}f(t_{i})-f(t_{i-1}){\bigg |}} where t i =

4331-449: The formula for general parametrizations, by considering the parametrization For a curve defined by an implicit equation F ( x , y ) = 0 with partial derivatives denoted F x , F y , F xx , F xy , F yy , the curvature is given by The signed curvature is not defined, as it depends on an orientation of the curve that is not provided by the implicit equation. Note that changing F into – F would not change

SECTION 60

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4402-461: The graph has a downward concavity. If it is zero, then one has an inflection point or an undulation point . When the slope of the graph (that is the derivative of the function) is small, the signed curvature is well approximated by the second derivative. More precisely, using big O notation , one has It is common in physics and engineering to approximate the curvature with the second derivative, for example, in beam theory or for deriving

4473-665: The left side of < {\displaystyle <} approaches 0 {\displaystyle 0} . In other words, ∑ i = 1 N | f ( t i ) − f ( t i − 1 ) Δ t | Δ t = ∑ i = 1 N | f ′ ( t i ) | Δ t {\displaystyle \sum _{i=1}^{N}\left|{\frac {f(t_{i})-f(t_{i-1})}{\Delta t}}\right|\Delta t=\sum _{i=1}^{N}\left|f'(t_{i})\right|\Delta t} in this limit, and

4544-806: The leftmost side | f ′ ( t i ) | = ∫ 0 1 | f ′ ( t i ) | d θ {\textstyle \left|f'(t_{i})\right|=\int _{0}^{1}\left|f'(t_{i})\right|d\theta } is used. By | | f ′ ( t i − 1 + θ ( t i − t i − 1 ) ) | − | f ′ ( t i ) | | < ε {\textstyle \left|\left|f'(t_{i-1}+\theta (t_{i}-t_{i-1}))\right|-\left|f'(t_{i})\right|\right|<\varepsilon } for N > ( b −

4615-425: The length of a quarter of the unit circle is ∫ − 2 / 2 2 / 2 d x 1 − x 2 . {\displaystyle \int _{-{\sqrt {2}}/2}^{{\sqrt {2}}/2}{\frac {dx}{\sqrt {1-x^{2}}}}\,.} The 15-point Gauss–Kronrod rule estimate for this integral of 1.570 796 326 808 177 differs from

4686-402: The numerator and the denominator in the preceding formula. Arc length Arc length is the distance between two points along a section of a curve . Determining the length of an irregular arc segment by approximating the arc segment as connected (straight) line segments is also called curve rectification . For a rectifiable curve these approximations don't get arbitrarily large (so

4757-452: The right side of this equality is just the Riemann integral of | f ′ ( t ) | {\displaystyle \left|f'(t)\right|} on [ a , b ] . {\displaystyle [a,b].} This definition of arc length shows that the length of a curve represented by a continuously differentiable function f : [

4828-937: The squared integrand of the arc length integral is ( x r ⋅ x r ) ( r ′ ) 2 + 2 ( x r ⋅ x θ ) r ′ θ ′ + ( x θ ⋅ x θ ) ( θ ′ ) 2 = ( r ′ ) 2 + r 2 ( θ ′ ) 2 . {\displaystyle \left(\mathbf {x_{r}} \cdot \mathbf {x_{r}} \right)\left(r'\right)^{2}+2\left(\mathbf {x} _{r}\cdot \mathbf {x} _{\theta }\right)r'\theta '+\left(\mathbf {x} _{\theta }\cdot \mathbf {x} _{\theta }\right)\left(\theta '\right)^{2}=\left(r'\right)^{2}+r^{2}\left(\theta '\right)^{2}.} So for

4899-1110: The squared norm of this vector is ( x r ⋅ x r ) ( r ′ 2 ) + ( x θ ⋅ x θ ) ( θ ′ ) 2 + ( x ϕ ⋅ x ϕ ) ( ϕ ′ ) 2 = ( r ′ ) 2 + r 2 ( θ ′ ) 2 + r 2 sin 2 ⁡ θ ( ϕ ′ ) 2 . {\displaystyle \left(\mathbf {x} _{r}\cdot \mathbf {x} _{r}\right)\left(r'^{2}\right)+\left(\mathbf {x} _{\theta }\cdot \mathbf {x} _{\theta }\right)\left(\theta '\right)^{2}+\left(\mathbf {x} _{\phi }\cdot \mathbf {x} _{\phi }\right)\left(\phi '\right)^{2}=\left(r'\right)^{2}+r^{2}\left(\theta '\right)^{2}+r^{2}\sin ^{2}\theta \left(\phi '\right)^{2}.} So for

4970-402: The tangent vector has a length equal to one and is thus a unit tangent vector . If the curve is twice differentiable, that is, if the second derivatives of x and y exist, then the derivative of T ( s ) exists. This vector is normal to the curve, its length is the curvature κ ( s ) , and it is oriented toward the center of curvature. That is, Moreover, because the radius of curvature

5041-446: The true length of arcsin ⁡ x | − 2 / 2 2 / 2 = π 2 {\displaystyle \arcsin x{\bigg |}_{-{\sqrt {2}}/2}^{{\sqrt {2}}/2}={\frac {\pi }{2}}} by 1.3 × 10 and the 16-point Gaussian quadrature rule estimate of 1.570 796 326 794 727 differs from the true length by only 1.7 × 10 . This means it

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