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Within differential calculus, in many applications, one needs to calculate the rate of change of a volume or surface integral whose domain of integration, as well as the integrand, are functions of a particular parameter. In physical applications, that parameter is frequently time *t*.

The rate of change of one-dimensional integrals with sufficiently smooth integrands, is governed by this extension of the fundamental theorem of calculus:

The calculus of moving surfaces ^{ [1] } provides analogous formulas for volume integrals over Euclidean domains, and surface integrals over differential geometry of surfaces, curved surfaces, including integrals over curved surfaces with moving contour boundaries.

Let *t* be a time-like parameter and consider a time-dependent domain Ω with a smooth surface boundary *S*. Let *F* be a time-dependent invariant field defined in the interior of Ω. Then the rate of change of the integral

is governed by the following law:^{ [1] }

where *C* is the velocity of the interface. The velocity of the interface *C* is the fundamental concept in the calculus of moving surfaces. In the above equation, *C* must be expressed with respect to the exterior normal. This law can be considered as the generalization of the fundamental theorem of calculus.

A related law governs the rate of change of the surface integral

The law reads

where the -derivative is the fundamental operator in the calculus of moving surfaces, originally proposed by Jacques Hadamard. is the trace of the mean curvature tensor. In this law, *C* need not be expression with respect to the exterior normal, as long as the choice of the normal is consistent for *C* and . The first term in the above equation captures the rate of change in *F* while the second corrects for expanding or shrinking area. The fact that mean curvature represents the rate of change in area follows from applying the above equation to since is area:

The above equation shows that mean curvature can be appropriately called the *shape gradient* of area. An evolution governed by

is the popular mean curvature flow and represents steepest descent with respect to area. Note that for a sphere of radius *R*, , and for a circle of radius *R*, with respect to the exterior normal.

Suppose that *S* is a moving surface with a moving contour γ. Suppose that the velocity of the contour γ with respect to *S* is *c*. Then the rate of change of the time dependent integral:

is

The last term captures the change in area due to annexation, as the figure on the right illustrates.

A **centripetal force** is a force that makes a body follow a curved path. Its direction is always orthogonal to the motion of the body and towards the fixed point of the instantaneous center of curvature of the path. Isaac Newton described it as "a force by which bodies are drawn or impelled, or in any way tend, towards a point as to a centre". In Newtonian mechanics, gravity provides the centripetal force causing astronomical orbits.

In mathematics, the **Laplace transform**, named after its inventor Pierre-Simon Laplace, is an integral transform that converts a function of a real variable to a function of a complex variable . The transform has many applications in science and engineering because it is a tool for solving differential equations. In particular, it transforms linear differential equations into algebraic equations and convolution into multiplication.

In vector calculus and differential geometry the **generalized Stokes theorem**, also called the **Stokes–Cartan theorem**, is a statement about the integration of differential forms on manifolds, which both simplifies and generalizes several theorems from vector calculus. It is a generalization of Isaac Newton's fundamental theorem of calculus that relates two-dimensional line integrals to three-dimensional surface integrals.

**Fractional calculus** is a branch of mathematical analysis that studies the several different possibilities of defining real number powers or complex number powers of the differentiation operator D

In physics, **Liouville's theorem**, named after the French mathematician Joseph Liouville, is a key theorem in classical statistical and Hamiltonian mechanics. It asserts that *the phase-space distribution function is constant along the trajectories of the system*—that is that the density of system points in the vicinity of a given system point traveling through phase-space is constant with time. This time-independent density is in statistical mechanics known as the classical a priori probability.

The **path integral formulation** is a description in quantum mechanics that generalizes the action principle of classical mechanics. It replaces the classical notion of a single, unique classical trajectory for a system with a sum, or functional integral, over an infinity of quantum-mechanically possible trajectories to compute a quantum amplitude.

In relativity, **proper time** along a timelike world line is defined as the time as measured by a clock following that line. It is thus independent of coordinates, and is a Lorentz scalar. The **proper time interval** between two events on a world line is the change in proper time. This interval is the quantity of interest, since proper time itself is fixed only up to an arbitrary additive constant, namely the setting of the clock at some event along the world line.

In physics, the **Hamilton–Jacobi equation**, named after William Rowan Hamilton and Carl Gustav Jacob Jacobi, is an alternative formulation of classical mechanics, equivalent to other formulations such as Newton's laws of motion, Lagrangian mechanics and Hamiltonian mechanics. The Hamilton–Jacobi equation is particularly useful in identifying conserved quantities for mechanical systems, which may be possible even when the mechanical problem itself cannot be solved completely.

In differential geometry, the **four-gradient** is the four-vector analogue of the gradient from vector calculus.

In general relativity, the **Gibbons–Hawking–York boundary term** is a term that needs to be added to the Einstein–Hilbert action when the underlying spacetime manifold has a boundary.

In calculus, the **Leibniz integral rule** for differentiation under the integral sign, named after Gottfried Leibniz, states that for an integral of the form

A **theoretical motivation for general relativity**, including the motivation for the geodesic equation and the Einstein field equation, can be obtained from special relativity by examining the dynamics of particles in circular orbits about the earth. A key advantage in examining circular orbits is that it is possible to know the solution of the Einstein Field Equation *a priori*. This provides a means to inform and verify the formalism.

In physics, **Maxwell's equations in curved spacetime** govern the dynamics of the electromagnetic field in curved spacetime or where one uses an arbitrary coordinate system. These equations can be viewed as a generalization of the vacuum Maxwell's equations which are normally formulated in the local coordinates of flat spacetime. But because general relativity dictates that the presence of electromagnetic fields induce curvature in spacetime, Maxwell's equations in flat spacetime should be viewed as a convenient approximation.

In probability theory and statistics, the **skew normal distribution** is a continuous probability distribution that generalises the normal distribution to allow for non-zero skewness.

The **fundamental theorem of calculus** is a theorem that links the concept of differentiating a function with the concept of integrating a function. The two operations are inverses of each other apart from a constant value which is dependent on where one starts to compute area.

In continuum mechanics, a **compatible** deformation **tensor field** in a body is that *unique* tensor field that is obtained when the body is subjected to a continuous, single-valued, displacement field. **Compatibility** is the study of the conditions under which such a displacement field can be guaranteed. Compatibility conditions are particular cases of integrability conditions and were first derived for linear elasticity by Barré de Saint-Venant in 1864 and proved rigorously by Beltrami in 1886.

The **vibration of plates** is a special case of the more general problem of mechanical vibrations. The equations governing the motion of plates are simpler than those for general three-dimensional objects because one of the dimensions of a plate is much smaller than the other two. This suggests that a two-dimensional plate theory will give an excellent approximation to the actual three-dimensional motion of a plate-like object, and indeed that is found to be true.

**Vasiliev equations** are *formally* consistent gauge invariant nonlinear equations whose linearization over a specific vacuum solution describes free massless higher-spin fields on anti-de Sitter space. The Vasiliev equations are classical equations and no Lagrangian is known that starts from canonical two-derivative Frønsdal Lagrangian and is completed by interactions terms. There is a number of variations of Vasiliev equations that work in three, four and arbitrary number of space-time dimensions. Vasiliev's equations admit supersymmetric extensions with any number of super-symmetries and allow for Yang-Mills gaugings. Vasiliev's equations are background independent, the simplest exact solution being anti-de Sitter space. It is important to note that locality is not properly implemented and the equations give a solution of certain formal deformation procedure, which is difficult to map to field theory language. The higher-spin AdS/CFT correspondence is reviewed in Higher-spin theory article.

In mathematics, **Katugampola fractional operators** are integral operators that generalize the *Riemann–Liouville* and the *Hadamard* fractional operators into a unique form. The **Katugampola fractional integral** generalizes both the Riemann–Liouville fractional integral and the Hadamard fractional integral into a single form and It is also closely related to the Erdelyi–Kober operator that generalizes the Riemann–Liouville fractional integral. **Katugampola fractional derivative** has been defined using the Katugampola fractional integral and as with any other fractional differential operator, it also extends the possibility of taking real number powers or complex number powers of the integral and differential operators.

In mathematics, **differential forms on a Riemann surface** are an important special case of the general theory of differential forms on smooth manifolds, distinguished by the fact that the conformal structure on the Riemann surface intrinsically defines a Hodge star operator on 1-forms without specifying a Riemannian metric. This allows the use of Hilbert space techniques for studying function theory on the Riemann surface and in particular for the construction of harmonic and holomorphic differentials with prescribed singularities. These methods were first used by Hilbert (1909) in his variational approach to the Dirichlet principle, making rigorous the arguments proposed by Riemann. Later Weyl (1940) found a direct approach using his method of orthogonal projection, a precursor of the modern theory of elliptic differential operators and Sobolev spaces. These techniques were originally applied to prove the uniformization theorem and its generalization to planar Riemann surfaces. Later they supplied the analytic foundations for the harmonic integrals of Hodge (1940). This article covers general results on differential forms on a Riemann surface that do not rely on any choice of Riemannian structure.

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