A **time derivative** is a derivative of a function with respect to time, usually interpreted as the rate of change of the value of the function.^{ [1] } The variable denoting time is usually written as .

A variety of notations are used to denote the time derivative. In addition to the normal (Leibniz's) notation,

A very common short-hand notation used, especially in physics, is the 'over-dot'. I.E.

(This is called Newton's notation)

Higher time derivatives are also used: the second derivative with respect to time is written as

with the corresponding shorthand of .

As a generalization, the time derivative of a vector, say:

is defined as the vector whose components are the derivatives of the components of the original vector. That is,

Time derivatives are a key concept in physics. For example, for a changing position , its time derivative is its velocity, and its second derivative with respect to time, , is its acceleration. Even higher derivatives are sometimes also used: the third derivative of position with respect to time is known as the jerk. See motion graphs and derivatives.

A large number of fundamental equations in physics involve first or second time derivatives of quantities. Many other fundamental quantities in science are time derivatives of one another:

- force is the time derivative of momentum
- power is the time derivative of energy
- electric current is the time derivative of electric charge

and so on.

A common occurrence in physics is the time derivative of a vector, such as velocity or displacement. In dealing with such a derivative, both magnitude and orientation may depend upon time.

For example, consider a particle moving in a circular path. Its position is given by the displacement vector , related to the angle, *θ*, and radial distance, *r*, as defined in the figure:

For this example, we assume that *θ* = *t*. Hence, the displacement (position) at any time *t* is given by

This form shows the motion described by **r**(*t*) is in a circle of radius *r* because the *magnitude* of **r**(*t*) is given by

using the trigonometric identity sin^{2}(*t*) + cos^{2}(*t*) = 1 and where is the usual Euclidean dot product.

With this form for the displacement, the velocity now is found. The time derivative of the displacement vector is the velocity vector. In general, the derivative of a vector is a vector made up of components each of which is the derivative of the corresponding component of the original vector. Thus, in this case, the velocity vector is:

Thus the velocity of the particle is nonzero even though the magnitude of the position (that is, the radius of the path) is constant. The velocity is directed perpendicular to the displacement, as can be established using the dot product:

Acceleration is then the time-derivative of velocity:

The acceleration is directed inward, toward the axis of rotation. It points opposite to the position vector and perpendicular to the velocity vector. This inward-directed acceleration is called centripetal acceleration.

In differential geometry, quantities are often expressed with respect to the local covariant basis, , where *i* ranges over the number of dimensions. The components of a vector expressed this way transform as a contravariant tensor, as shown in the expression , invoking Einstein summation convention. If we want to calculate the time derivatives of these components along a trajectory, so that we have , we can define a new operator, the invariant derivative , which will continue to return contravariant tensors:^{ [2] }

where (with being the *j*th coordinate) captures the components of the velocity in the local covariant basis, and are the Christoffel symbols for the coordinate system. Note that explicit dependence on *t* has been repressed in the notation. We can then write:

as well as:

In terms of the covariant derivative, , we have:

In economics, many theoretical models of the evolution of various economic variables are constructed in continuous time and therefore employ time derivatives.^{ [3] }^{: ch. 1-3 } One situation involves a stock variable and its time derivative, a flow variable. Examples include:

- The flow of net fixed investment is the time derivative of the capital stock.
- The flow of inventory investment is the time derivative of the stock of inventories.
- The growth rate of the money supply is the time derivative of the money supply divided by the money supply itself.

Sometimes the time derivative of a flow variable can appear in a model:

- The growth rate of output is the time derivative of the flow of output divided by output itself.
- The growth rate of the labor force is the time derivative of the labor force divided by the labor force itself.

And sometimes there appears a time derivative of a variable which, unlike the examples above, is not measured in units of currency:

- The time derivative of a key interest rate can appear.
- The inflation rate is the growth rate of the price level—that is, the time derivative of the price level divided by the price level itself.

In mechanics, **acceleration** is the rate of change of the velocity of an object with respect to time. Accelerations are vector quantities. The orientation of an object's acceleration is given by the orientation of the *net* force acting on that object. The magnitude of an object's acceleration, as described by Newton's Second Law, is the combined effect of two causes:

In calculus, the **chain rule** is a formula that expresses the derivative of the composition of two differentiable functions f and g in terms of the derivatives of f and g. More precisely, if is the function such that for every x, then the chain rule is, in Lagrange's notation,

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 **derivative** of a function of a real variable measures the sensitivity to change of the function value with respect to a change in its argument. Derivatives are a fundamental tool of calculus. For example, the derivative of the position of a moving object with respect to time is the object's velocity: this measures how quickly the position of the object changes when time advances.

In celestial mechanics, an **orbit** is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an artificial satellite around an object or position in space such as a planet, moon, asteroid, or Lagrange point. Normally, orbit refers to a regularly repeating trajectory, although it may also refer to a non-repeating trajectory. To a close approximation, planets and satellites follow elliptic orbits, with the center of mass being orbited at a focal point of the ellipse, as described by Kepler's laws of planetary motion.

In physics, the **Navier–Stokes equations** are certain partial differential equations which describe the motion of viscous fluid substances, named after French engineer and physicist Claude-Louis Navier and Anglo-Irish physicist and mathematician George Gabriel Stokes. They were developed over several decades of progressively building the theories, from 1822 (Navier) to 1842–1850 (Stokes).

In physics, **equations of motion** are equations that describe the behavior of a physical system in terms of its motion as a function of time. More specifically, the equations of motion describe the behavior of a physical system as a set of mathematical functions in terms of dynamic variables. These variables are usually spatial coordinates and time, but may include momentum components. The most general choice are generalized coordinates which can be any convenient variables characteristic of the physical system. The functions are defined in a Euclidean space in classical mechanics, but are replaced by curved spaces in relativity. If the dynamics of a system is known, the equations are the solutions for the differential equations describing the motion of the dynamics.

**Kinematics** is a subfield of physics, developed in classical mechanics, that describes the motion of points, bodies (objects), and systems of bodies without considering the forces that cause them to move. Kinematics, as a field of study, is often referred to as the "geometry of motion" and is occasionally seen as a branch of mathematics. A kinematics problem begins by describing the geometry of the system and declaring the initial conditions of any known values of position, velocity and/or acceleration of points within the system. Then, using arguments from geometry, the position, velocity and acceleration of any unknown parts of the system can be determined. The study of how forces act on bodies falls within kinetics, not kinematics. For further details, see analytical dynamics.

In physics, **work** is the energy transferred to or from an object via the application of force along a displacement. In its simplest form, for a constant force aligned with the direction of motion, it equals the product of the force strength and the distance traveled. A force is said to do *positive work* if when applied it has a component in the direction of the displacement of the point of application. A force does *negative work* if it has a component opposite to the direction of the displacement at the point of application of the force.

In mathematics, a **unit vector** in a normed vector space is a vector of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or "hat", as in .

In mathematics, the **Laplace operator** or **Laplacian** is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. It is usually denoted by the symbols , , or . In a Cartesian coordinate system, the Laplacian is given by the sum of second partial derivatives of the function with respect to each independent variable. In other coordinate systems, such as cylindrical and spherical coordinates, the Laplacian also has a useful form. Informally, the Laplacian Δ*f* (*p*) of a function *f* at a point *p* measures by how much the average value of *f* over small spheres or balls centered at *p* deviates from *f* (*p*).

In special relativity, a **four-vector** is an object with four components, which transform in a specific way under Lorentz transformations. Specifically, a four-vector is an element of a four-dimensional vector space considered as a representation space of the standard representation of the Lorentz group, the representation. It differs from a Euclidean vector in how its magnitude is determined. The transformations that preserve this magnitude are the Lorentz transformations, which include spatial rotations and boosts.

In analytical mechanics, **generalized coordinates** are a set of parameters used to represent the state of a system in a configuration space. These parameters must uniquely define the configuration of the system relative to a reference state. The **generalized velocities** are the time derivatives of the generalized coordinates of the system. The adjective "generalized" distinguishes these parameters from the traditional use of the term "coordinate" to refer to Cartesian coordinates

In physics, **circular motion** is a movement of an object along the circumference of a circle or rotation along a circular path. It can be uniform, with constant angular rate of rotation and constant speed, or non-uniform with a changing rate of rotation. The rotation around a fixed axis of a three-dimensional body involves circular motion of its parts. The equations of motion describe the movement of the center of mass of a body. In circular motion, the distance between the body and a fixed point on the surface remains the same.

A **fictitious force** is a force that appears to act on a mass whose motion is described using a non-inertial frame of reference, such as a linearly accelerating or rotating reference frame. It is related to Newton's second law of motion, which treats forces for just one object.

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

**Linear motion**, also called **rectilinear motion**, is one-dimensional motion along a straight line, and can therefore be described mathematically using only one spatial dimension. The linear motion can be of two types: uniform linear motion with constant velocity or zero acceleration; and non-uniform linear motion with variable velocity or non-zero acceleration. The motion of a particle along a line can be described by its position , which varies with (time). An example of linear motion is an athlete running 100m along a straight track.

In fluid dynamics, the **Oseen equations** describe the flow of a viscous and incompressible fluid at small Reynolds numbers, as formulated by Carl Wilhelm Oseen in 1910. Oseen flow is an improved description of these flows, as compared to Stokes flow, with the (partial) inclusion of convective acceleration.

In physics, **Lagrangian mechanics** is a formulation of classical mechanics founded on the stationary-action principle. It was introduced by the Italian-French mathematician and astronomer Joseph-Louis Lagrange in his 1788 work, *Mécanique analytique*.

In classical mechanics, the **central-force problem** is to determine the motion of a particle in a single central potential field. A central force is a force that points from the particle directly towards a fixed point in space, the center, and whose magnitude only depends on the distance of the object to the center. In a few important cases, the problem can be solved analytically, i.e., in terms of well-studied functions such as trigonometric functions.

- ↑ Chiang, Alpha C.,
*Fundamental Methods of Mathematical Economics*, McGraw-Hill, third edition, 1984, ch. 14, 15, 18. - ↑ Grinfeld, Pavel. "Tensor Calculus 6d: Velocity, Acceleration, Jolt and the New δ/δt-derivative".
*YouTube*. Archived from the original on 2021-12-13. - ↑ See for example Romer, David (1996).
*Advanced Macroeconomics*. McGraw-Hill. ISBN 0-07-053667-8.

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