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Integration using parametric derivatives information


In calculus, integration by parametric derivatives, also called parametric integration,[1] is a method which uses known Integrals to integrate derived functions. It is often used in Physics, and is similar to integration by substitution.

  1. ^ Zatja, Aurel J. (December 1989). "Parametric Integration Techniques | Mathematical Association of America" (PDF). www.maa.org. Mathematics Magazine. Retrieved 23 July 2019.

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Integration using parametric derivatives

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Derivative

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Parametric surface

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curve on the surface and the surface area can be found using integration. Let the parametric surface be given by the equation r = r ( u , v ) , {\displaystyle...

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differentiable function, that is, a function that has derivatives of all orders (this implies that all these derivatives are continuous). Generally, the term smooth...

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x is a dummy variable or variable of integration (confusingly, also sometimes called a parameter of integration). In statistics and econometrics, the...

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calculus, and more generally in mathematical analysis, integration by parts or partial integration is a process that finds the integral of a product of...

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Multivariable calculus

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curl in terms of partial derivatives. A matrix of partial derivatives, the Jacobian matrix, may be used to represent the derivative of a function between...

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Matrix calculus

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will find that the derivatives involving matrices will reduce to derivatives involving vectors in a corresponding way. The derivative of a vector function...

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series Parametric equations Polar functions (including arc length in polar coordinates and calculating area) Arc length calculations using integration Integration...

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the derivative, one repeatedly applies partial derivatives with respect to different variables. For example, the second order partial derivatives of a...

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Sound from ultrasound

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interaction of an aimed beam of ultrasound waves produced by a parametric array using heterodyning. Ultrasound has much shorter wavelengths than audible...

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Curvilinear coordinates

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_{z}={\dfrac {\partial \mathbf {r} }{\partial z}}.} Applying the same derivatives to the curvilinear system locally at point P defines the natural basis...

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Computer algebra system

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Landau's algorithm (nested radicals) Derivatives of elementary functions and special functions. (e.g. See derivatives of the incomplete gamma function.)...

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Solid of revolution

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orders of integration. The disc method is used when the slice that was drawn is perpendicular to the axis of revolution; i.e. when integrating parallel...

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Surface area

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on methods of infinitesimal calculus and involves partial derivatives and double integration. A general definition of surface area was sought by Henri...

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Common integrals in quantum field theory

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This decouples the variables and allows the integration to be performed as n one-dimensional integrations. This is best illustrated with a two-dimensional...

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Gradient

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partial derivatives exist in every direction but fail to be differentiable. Furthermore, this definition as the vector of partial derivatives is only...

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Desmos

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the 2D calculator, including cross products, partial derivatives and double-variable parametric equations. In addition to graphing both equations and...

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Kinematics

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a Δ r . {\displaystyle v^{2}=v_{0}^{2}+2a\Delta r.} This reduces the parametric equations of motion of the particle to a Cartesian relationship of speed...

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integrals involving functions and their derivatives. Functions that maximize or minimize functionals may be found using the Euler–Lagrange equation of the...

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Precalculus

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and trigonometric identities. The binomial theorem, polar coordinates, parametric equations, and the limits of sequences and series are other common topics...

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response Zhang, Weihong; Zhou, Ying (2021). "Level-set functions and parametric functions". The Feature-Driven Method for Structural Optimization. Elsevier...

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Beltrami identity

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Ellipse

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\end{aligned}}} where atan2 is the 2-argument arctangent function. Using trigonometric functions, a parametric representation of the standard ellipse x 2 a 2 + y 2...

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{1}{2}}\sigma ^{2}K^{2}C_{KK}\end{aligned}}} Dupire's approach is non-parametric. It requires to pre-interpolate the data to obtain a continuum of traded...

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non-parametrically, or semi-parametrically, simply as 'smooth functions', to be estimated by non-parametric means. So a typical GAM might use a scatterplot smoothing...

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closed-form solutions for arc length and numerical integration is necessary. Numerical integration of the arc length integral is usually very efficient...

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