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| {{Calculus |Multivariable}}
| | They're always ready to help, and they're always making changes to the site to make sure you won't have troubles in the first place. Medical word press themes give you the latest medical designs. These templates are professionally designed and are also Adsense ready. Transforming your designs to Word - Press blogs is not that easy because of the simplified way in creating your very own themes. Also our developers are well convergent with the latest technologies and bitty-gritty of wordpress website design and promises to deliver you the best solution that you can ever have. <br><br>Choosing what kind of links you'll be using is a ctitical aspect of any linkwheel strategy, especially since there are several different types of links that are assessed by search engines. WPTouch is among the more well known Word - Press smartphone plugins which is currently in use by thousands of users. Well Managed Administration The Word - Press can easily absorb the high numbers of traffic by controlling the server load to make sure that the site works properly. By purchasing Word - Press weblogs you can acquire your very own domain title and have total command of your web site. By using Word - Press, you can develop very rich, user-friendly and full-functional website. <br><br>Here are a few reasons as to why people prefer Word - Press over other software's. When a business benefits from its own domain name and a tailor-made blog, the odds of ranking higher in the search engines and being visible to a greater number of people is more likely. Whether or not it's an viewers on your web page, your social media pages, or your web page, those who have a present and effective viewers of "fans" are best best for provide provides, reductions, and deals to help re-invigorate their viewers and add to their main point here. Our skilled expertise, skillfulness and excellence have been well known all across the world. If you have any type of concerns regarding where and how to utilize [http://www.nt-protect.host.sk/?wordpress_backup_420779 wordpress dropbox backup], you could call us at the web-page. Have you heard about niche marketing and advertising. <br><br>Additionally Word - Press add a default theme named Twenty Fourteen. In case you need to hire PHP developers or hire Offshore Code - Igniter development services or you are looking for Word - Press development experts then Mindfire Solutions would be the right choice for a Software Development partner. Specialty about our themes are that they are easy to load, compatible with latest wordpress version and are also SEO friendly. Word - Press is the most popular open source content management system (CMS) in the world today. This includes enriching the content with proper key words, tactfully defining the tags and URL. <br><br>Internet is not only the source for information, it is also one of the source for passive income. Visit our website to learn more about how you can benefit. Offshore Wordpress development services from a legitimate source caters dedicated and professional services assistance with very simplified yet technically effective development and designing techniques from experienced professional Wordpress developer India. In addition, Word - Press design integration is also possible. The 2010 voting took place from July 7 through August 31, 2010. |
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| In [[mathematics]], a '''line integral''' (sometimes called a '''path integral''', '''[[contour integral]]''', '''curve integral''', or '''curvilinear integral'''; not to be confused with [[arc length|calculating arc length using integration]]) is an [[integral]] where the [[function (mathematics)|function]] to be integrated is evaluated along a [[curve]].
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| The function to be integrated may be a [[scalar field]] or a [[vector field]]. The value of the line integral is the sum of values of the field at all points on the curve, weighted by some scalar function on the curve (commonly [[arc length]] or, for a vector field, the [[Dot product|scalar product]] of the vector field with a [[Differential (infinitesimal)|differential]] vector in the curve). This weighting distinguishes the line integral from simpler integrals defined on [[interval (mathematics)|intervals]]. Many simple formulae in physics (for [[Mechanical work|example]], ''W''='''F'''·'''s''') have natural continuous analogs in terms of line integrals (''W''=∫<sub>''C''</sub> '''F'''· d'''s'''). The line integral finds the [[Mechanical work|work]] done on an object moving through an electric or gravitational field, for example.
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| == Vector calculus ==
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| In qualitative terms, a line integral in vector calculus can be thought of as a measure of the total effect of a given [[Field (physics)|field]] along a given curve. More specifically, the line integral over a scalar field can be interpreted as the area under the field carved out by a particular curve. This can be visualized as the surface created by ''z'' = ''f''(''x'',''y'') and a curve ''C'' in the ''x''-''y'' plane. The line integral of ''f'' would be the area of the "curtain" created when the points of the surface that are directly over ''C'' are carved out.
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| === Line integral of a scalar field ===
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| [[File:Line integral of scalar field.gif|frame|right|The line integral over a scalar field ''f'' can be thought of as the area under the curve ''C'' along a surface ''z'' = ''f''(''x'',''y''), described by the field.]]
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| ==== Definition ====
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| For some [[scalar field]] ''f'' : ''U'' ⊆ '''R'''<sup>''n''</sup> → '''R''', the line integral along a [[piecewise smooth]] [[curve]] ''C'' ⊂ ''U'' is defined as
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| :<math>\int\limits_C f\, ds = \int_a^b f(\mathbf{r}(t)) |\mathbf{r}'(t)|\, dt.</math>
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| where '''r''': [a, b] → ''C'' is an arbitrary [[bijective]] [[parametric equation|parametrization]] of the curve ''C'' such that '''r'''(''a'') and '''r'''(''b'') give the endpoints of ''C'' and <math>a < b</math>.
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| The function ''f'' is called the integrand, the curve ''C'' is the domain of integration, and the symbol ''ds'' may be intuitively interpreted as an elementary [[arc length]]. Line integrals of scalar fields over a curve ''C'' do not depend on the chosen parametrization '''r''' of ''C''.
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| Geometrically, when the scalar field ''f'' is defined over a plane (''n''=2), its graph is a surface ''z''=''f''(''x'',''y'') in space, and the line integral gives the (signed) [[Cross-section (geometry)|cross-sectional]] area bounded by the curve ''C'' and the graph of ''f''. See the animation to the right.
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| ==== Derivation ====
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| For a '''line integral''' over a scalar field, the integral can be constructed from a [[Riemann sum]] using the above definitions of ''f'', ''C'' and a parametrization '''r''' of ''C''. This can be done by partitioning the [[interval (mathematics)|interval]] [''a'',''b''] into ''n'' sub-intervals [''t''<sub>''i-1''</sub>, ''t''<sub>''i''</sub>] of length Δ''t'' = (''b'' − ''a'')/''n'', then '''r'''(''t''<sub>''i''</sub>) denotes some point, call it a sample point, on the curve ''C''. We can use the set of sample points {'''r'''(''t''<sub>''i''</sub>) : 1 ≤ i ≤ n} to approximate the curve ''C'' by a [[polygonal path]] by introducing a straight line piece between each of the sample points '''r'''(''t''<sub>''i-1''</sub>) and '''r'''(''t''<sub>''i''</sub>). We then label the distance between each of the sample points on the curve as Δ''s''<sub>''i''</sub>. The product of ''f''('''r'''(''t''<sub>''i''</sub>)) and Δ''s''<sub>''i''</sub> can be associated with the signed area of a rectangle with a height and width of ''f''('''r'''(''t''<sub>''i''</sub>)) and Δ''s''<sub>''i''</sub> respectively. Taking the [[Limit of a sequence|limit]] of the [[sum]] of the terms as the length of the partitions approaches zero gives us <math>I</math>
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| :<math>I = \lim_{\Delta s_i \rightarrow 0} \sum_{i=1}^n f(\mathbf{r}(t_i))\Delta s_i.</math>
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| We note that, by the [[mean value theorem]], the distance between subsequent points on the curve, is
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| :<math>\Delta s_i = |\mathbf{r}(t_i+\Delta t)-\mathbf{r}(t_i)|\approx|\mathbf{r}'(t_i)|\Delta t.</math>
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| Substituting this in the above Riemann sum yields
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| :<math>I = \lim_{\Delta t \rightarrow 0} \sum_{i=1}^n f(\mathbf{r}(t_i))|\mathbf{r}'(t_i)|\Delta t</math>
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| which is the Riemann sum for the integral
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| :<math>I = \int_a^b f(\mathbf{r}(t)) |\mathbf{r}'(t)|\, dt.</math>
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| === Line integral of a vector field ===
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| ==== Definition ====
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| For a [[vector field]] '''F''' : ''U'' ⊆ '''R'''<sup>''n''</sup> → '''R'''<sup>''n''</sup>, the line integral along a [[piecewise smooth]] [[curve]] ''C'' ⊂ ''U'', in the direction of '''r''', is defined as
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| :<math>\int\limits_C \mathbf{F}(\mathbf{r})\cdot\,d\mathbf{r} = \int_a^b \mathbf{F}(\mathbf{r}(t))\cdot\mathbf{r}'(t)\,dt.</math>
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| where · is the [[dot product]] and '''r''': [a, b] → ''C'' is a [[bijective]] [[parametric equation|parametrization]] of the curve ''C'' such that '''r'''(''a'') and '''r'''(''b'') give the endpoints of ''C''.
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| A line integral of a scalar field is thus a line integral of a vector field where the vectors are always [[tangential]] to the line.
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| Line integrals of vector fields are independent of the parametrization '''r''' in [[absolute value]], but they do depend on its [[curve orientation|orientation]]. Specifically, a reversal in the orientation of the parametrization changes the sign of the line integral.
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| The line integral of a vector field along a curve is the integral of the corresponding 1-form under the [[musical isomorphism]] over the curve considered as an immersed
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| 1-manifold.
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| ==== Derivation ====
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| [[Image:Line integral of vector field.gif|450px|thumb|right|The trajectory of a particle (in red) along a curve inside a vector field. Starting from a, the particle traces the path C along the vector field F. The dot product (green line) of its displacement vector (red arrow) and the field vector (blue arrow) defines an area under a curve, which is equivalent to the path's line integral. (Click on image for a detailed description.)]]
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| The line integral of a vector field can be derived in a manner very similar to the case of a scalar field. Again using the above definitions of '''F''', ''C'' and its parametrization '''r'''(''t''), we construct the integral from a [[Riemann sum]]. Partition the interval [''a'',''b''] into ''n'' intervals of length Δ''t'' = (''b'' − ''a'')/''n''. Letting ''t<sub>i</sub>'' be the ''i''th point on [''a'',''b''], then '''r'''(''t''<sub>''i''</sub>) gives us the position of the ''i''th point on the curve. However, instead of calculating up the distances between subsequent points, we need to calculate their [[displacement (vector)|displacement]] vectors, Δ'''r'''<sub>''i''</sub>. As before, evaluating '''F''' at all the points on the curve and taking the dot product with each displacement vector gives us the [[infinitesimal]] contribution of each partition of '''F''' on ''C''. Letting the size of the partitions go to zero gives us a sum
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| :<math>I = \lim_{\Delta t \rightarrow 0} \sum_{i=1}^n \mathbf{F}(\mathbf{r}(t_i)) \cdot \Delta\mathbf{r}_i</math>
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| By the [[mean value theorem]], we see that the displacement vector between adjacent points on the curve is
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| :<math>\Delta\mathbf{r}_i = \mathbf{r}(t_i+\Delta t)-\mathbf{r}(t_i)\approx\mathbf{r}'(t_i)\Delta t</math> | |
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| Substituting this in the above Riemann sum yields
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| :<math>I = \lim_{\Delta t \rightarrow 0} \sum_{i=1}^n \mathbf{F}(\mathbf{r}(t_i)) \cdot \mathbf{r}'(t_i)\Delta t</math>
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| which is the Riemann sum for the integral defined above.
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| === Path independence ===
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| {{Main|Gradient theorem}}
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| If a vector field '''F''' is the [[gradient]] of a [[scalar field]] ''G'' (i.e. if '''F''' is conservative), that is,
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| :<math>\nabla G = \mathbf{F},</math>
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| then the [[derivative]] of the [[function composition|composition]] of ''G'' and '''r'''(''t'') is
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| :<math>\frac{dG(\mathbf{r}(t))}{dt} = \nabla G(\mathbf{r}(t)) \cdot \mathbf{r}'(t) = \mathbf{F}(\mathbf{r}(t)) \cdot \mathbf{r}'(t)</math>
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| which happens to be the integrand for the line integral of '''F''' on '''r'''(''t''). It follows that, given a path ''C '', then
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| :<math>\int_C \mathbf{F}(\mathbf{r})\cdot\,d\mathbf{r} = \int_a^b \mathbf{F}(\mathbf{r}(t))\cdot\mathbf{r}'(t)\,dt = \int_a^b \frac{dG(\mathbf{r}(t))}{dt}\,dt = G(\mathbf{r}(b)) - G(\mathbf{r}(a)).</math>
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| In other words, the integral of '''F''' over ''C'' depends solely on the values of ''G'' in the points '''r'''(''b'') and '''r'''(''a'') and is thus independent of the path between them.
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| For this reason, a line integral of a conservative vector field is called ''path independent''.
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| === Applications ===
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| The line integral has many uses in physics. For example, the [[Work (physics)|work]] done on a particle traveling on a curve ''C'' inside a force field represented as a vector field '''F''' is the line integral of '''F''' on ''C''.
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| == Complex line integral ==<!-- This section is linked from [[Complex plane]] -->
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| In [[complex analysis]], the line integral is defined in terms of [[Complex number#Multiplication and division|multiplication]] and [[Complex number#Addition and subtraction|addition]] of complex numbers. Suppose ''U'' is an [[open subset]] of the [[complex plane]] '''C''', ''f'' : ''U'' → '''C''' is a function, and <math>L\subset U</math> is a curve of finite length, parametrized by <math>\gamma : [a,b] \to L</math>, where <math> \gamma(t)=x(t)+iy(t).</math> The line integral
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| :<math>\int_L f(z)\,dz</math>
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| may be defined by subdividing the [[interval (mathematics)|interval]] [''a'', ''b''] into ''a'' = ''t''<sub>0</sub> < ''t''<sub>1</sub> < ... < ''t''<sub>''n''</sub> = ''b'' and considering the expression
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| :<math>\sum_{k=1}^{n} f(\gamma(t_k)) [ \gamma(t_k) - \gamma(t_{k-1}) ]
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| =\sum_{k=1}^{n} f(\gamma_k) \Delta\gamma_k.</math>
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| The integral is then the limit of this [[Riemann sum]] as the lengths of the subdivision intervals approach zero.
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| If the parametrization <math>\gamma</math> is [[Differentiable#Differentiability classes|continuously differentiable]], the line integral can be evaluated as an integral of a function of a real variable:
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| :<math>\int_L f(z)\,dz
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| =\int_a^b f(\gamma(t))\,\gamma\,'(t)\,dt.</math>
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| When <math>L</math> is a closed curve, that is, its initial and final points coincide, the notation
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| :<math>\oint_L f(z)\,dz</math>
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| is often used for the line integral of ''f'' along <math>L</math>.
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| A closed curve line integral is sometimes referred to as a cyclic integral in engineering applications.
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| The line integral with respect to the conjugate complex differential <math>\overline{dz}</math> is defined <ref>{{cite book|first1=Lars| last1=Ahlfors | title = Complex Analysis 2nd edition | page=103}}</ref> to be
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| :<math>\int_L f \overline{dz} := \overline{\int_L \overline{f} dz} = \int_a^b f(\gamma(t))\,\overline{\gamma'(t)}\,dt.</math>
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| The line integrals of complex functions can be evaluated using a number of techniques: the integral may be split into real and imaginary parts reducing the problem to that of evaluating two real-valued line integrals, the [[Cauchy integral formula]] may be used in other circumstances. If the line integral is a closed curve in a region where the function is [[Analytic function|analytic]] and containing no [[Mathematical singularity|singularities]], then the value of the integral is simply zero; this is a consequence of the [[Cauchy integral theorem]]. The [[residue theorem]] allows contour integrals to be used in the complex plane to find integrals of real-valued functions of a real variable (see [[residue theorem]] for an example).
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| === Example ===
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| Consider the function ''f''(''z'')=1/''z'', and let the contour ''L'' be the [[unit circle]] about 0, parametrized by γ(''t'')=''e''<sup>''it''</sup> with ''t'' in [0, 2π] (which generates the circle counterclockwise). Substituting, we find
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| :<math>
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| \begin{align}
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| \oint_L f(z)\,dz & = \int_0^{2\pi} {1\over e^{it}} ie^{it}\,dt = i\int_0^{2\pi} e^{-it}e^{it}\,dt \\
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| & =i\int_0^{2\pi}\,dt = i(2\pi-0)=2\pi i.
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| \end{align}
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| </math> | |
| Here we have used the fact that any complex number ''z'' can be written as ''re<sup>it</sup>'' where ''r'' is the [[Absolute value|modulus]] of ''z''. On the unit circle this is fixed to 1, so the only variable left is the angle, which is denoted by ''t''. This answer can be also verified by the [[Cauchy integral formula]].
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| === Relation between the line integral of a vector field and the complex line integral ===
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| Viewing complex numbers as 2-dimensional [[Euclidean vector|vectors]], the line integral of a 2-dimensional vector field corresponds to the real part of the line integral of the [[Complex conjugate|conjugate]] of the corresponding complex function of a complex variable. More specifically, if <math>\mathbf{r} (t) = (x(t), y(t))</math> is a parameterization of ''L'' and <math>f(z)=u(z)+iv(z)</math>, then:
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| :<math>\int_L \overline{f(z)}\,dz = \int_L \bar{f}\,dx + i\int_L \bar{f}\,dy = \int_L (u,v)\cdot d\mathbf{r} + i\int_L (-v,u)\cdot d\mathbf{r},</math>
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| provided that both integrals on the right hand side exist, and that the parametrization <math>\gamma</math> of ''L'' has the same orientation as <math>\mathbf{r}</math> (just expand the Riemann sum for the lefthand integral and take the limit).
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| By [[Green's theorem]], the area of a region enclosed by a smooth, closed, positively oriented curve <math>L</math> is given by the integral
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| :<math>\frac{1}{2i} \int_L \overline{z} \; dz</math>
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| This fact is used, for example, in the proof of the [[area theorem]].
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| Due to the [[Cauchy-Riemann equations]] the [[Curl (mathematics)|curl]] of the vector field corresponding to the conjugate of a [[holomorphic function]] is zero. This relates through [[Stokes' theorem]] both types of line integral being zero.
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| == Quantum mechanics ==
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| The "[[path integral formulation]]" of [[quantum mechanics]] actually refers not to path integrals in this sense but to [[functional integration|functional integrals]], that is, integrals over a space of paths, of a function ''of'' a possible path. However, path integrals in the sense of this article are important in quantum mechanics; for example, complex contour integration is often used in evaluating [[probability amplitude]]s in quantum [[scattering]] theory.
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| == References ==
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| {{Reflist|2}}
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| == See also ==
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| * [[Arc length]]
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| * [[Divergence theorem]]
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| * [[Functional integration]]
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| * [[Gradient theorem]]
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| * [[Green's theorem]]
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| * [[Methods of contour integration]]
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| * [[Nachbin's theorem]]
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| * [[Stokes' theorem]]
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| * [[Surface integral]]
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| * [[Volume element]]
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| * [[Volume integral]]
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| == External links ==
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| * {{springer|title=Integral over trajectories|id=p/i051570}}
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| * [[Khan Academy]] modules:
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| ** [http://www.khanacademy.org/video/introduction-to-the-line-integral?playlist=Calculus "Introduction to the Line Integral"]
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| ** [http://www.khanacademy.org/video/line-integral-example-1?playlist=Calculus "Line Integral Example 1"]
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| ** [http://www.khanacademy.org/video/line-integral-example-2--part-1?playlist=Calculus "Line Integral Example 2 (part 1)"]
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| ** [http://www.khanacademy.org/video/line-integral-example-2--part-2?playlist=Calculus "Line Integral Example 2 (part 2)"]
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| * {{PlanetMath|urlname=LineIntegral|title=Path integral}}
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| * [http://www.math.montana.edu/frankw/ccp/multiworld/traveling/line-integrals/body.htm A pictoral explanation of the path integral]
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| * [http://math.fullerton.edu/mathews/c2003/ContourIntegralMod.html Contour Integrals Module by John H. Mathews]
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| * [http://mathinsight.org/line_integral_vector_field_introduction Line integral of a vector field – Interactive]
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| [[Category:Complex analysis]]
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| [[Category:Vector calculus]]
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They're always ready to help, and they're always making changes to the site to make sure you won't have troubles in the first place. Medical word press themes give you the latest medical designs. These templates are professionally designed and are also Adsense ready. Transforming your designs to Word - Press blogs is not that easy because of the simplified way in creating your very own themes. Also our developers are well convergent with the latest technologies and bitty-gritty of wordpress website design and promises to deliver you the best solution that you can ever have.
Choosing what kind of links you'll be using is a ctitical aspect of any linkwheel strategy, especially since there are several different types of links that are assessed by search engines. WPTouch is among the more well known Word - Press smartphone plugins which is currently in use by thousands of users. Well Managed Administration The Word - Press can easily absorb the high numbers of traffic by controlling the server load to make sure that the site works properly. By purchasing Word - Press weblogs you can acquire your very own domain title and have total command of your web site. By using Word - Press, you can develop very rich, user-friendly and full-functional website.
Here are a few reasons as to why people prefer Word - Press over other software's. When a business benefits from its own domain name and a tailor-made blog, the odds of ranking higher in the search engines and being visible to a greater number of people is more likely. Whether or not it's an viewers on your web page, your social media pages, or your web page, those who have a present and effective viewers of "fans" are best best for provide provides, reductions, and deals to help re-invigorate their viewers and add to their main point here. Our skilled expertise, skillfulness and excellence have been well known all across the world. If you have any type of concerns regarding where and how to utilize wordpress dropbox backup, you could call us at the web-page. Have you heard about niche marketing and advertising.
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