Time Derivative
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A time derivative is a
derivative In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
of a function with respect to
time Time is the continued sequence of existence and events that occurs in an apparently irreversible succession from the past, through the present, into the future. It is a component quantity of various measurements used to sequence events, to ...
, usually interpreted as the rate of change of the value of the function. The variable denoting time is usually written as t.


Notation

A variety of notations are used to denote the time derivative. In addition to the normal ( Leibniz's) notation, :\frac A very common short-hand notation used, especially in physics, is the 'over-dot'. I.E. :\dot (This is called
Newton's notation In differential calculus, there is no single uniform notation for differentiation. Instead, various notations for the derivative of a function or variable have been proposed by various mathematicians. The usefulness of each notation varies with ...
) Higher time derivatives are also used: the
second derivative In calculus, the second derivative, or the second order derivative, of a function is the derivative of the derivative of . Roughly speaking, the second derivative measures how the rate of change of a quantity is itself changing; for example, ...
with respect to time is written as :\frac with the corresponding shorthand of \ddot. As a generalization, the time derivative of a vector, say: : \mathbf v = \left v_1,\ v_2,\ v_3, \ldots \right is defined as the vector whose components are the derivatives of the components of the original vector. That is, : \frac = \left \frac,\frac ,\frac , \ldots \right .


Use in physics

Time derivatives are a key concept in
physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
. For example, for a changing
position Position often refers to: * Position (geometry), the spatial location (rather than orientation) of an entity * Position, a job or occupation Position may also refer to: Games and recreation * Position (poker), location relative to the dealer * ...
x, its time derivative \dot is its
velocity Velocity is the directional speed of an object in motion as an indication of its rate of change in position as observed from a particular frame of reference and as measured by a particular standard of time (e.g. northbound). Velocity i ...
, and its second derivative with respect to time, \ddot, is its
acceleration In mechanics, acceleration is the rate of change of the velocity of an object with respect to time. Accelerations are vector quantities (in that they have magnitude and direction). The orientation of an object's acceleration is given by the ...
. 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 In physics, a force is an influence that can change the motion of an object. A force can cause an object with mass to change its velocity (e.g. moving from a state of rest), i.e., to accelerate. Force can also be described intuitively as a p ...
is the time derivative of
momentum In Newtonian mechanics, momentum (more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If is an object's mass an ...
*
power Power most often refers to: * Power (physics), meaning "rate of doing work" ** Engine power, the power put out by an engine ** Electric power * Power (social and political), the ability to influence people or events ** Abusive power Power may a ...
is the time derivative of
energy In physics, energy (from Ancient Greek: ἐνέργεια, ''enérgeia'', “activity”) is the quantitative property that is transferred to a body or to a physical system, recognizable in the performance of work and in the form of heat a ...
*
electric current An electric current is a stream of charged particles, such as electrons or ions, moving through an electrical conductor or space. It is measured as the net rate of flow of electric charge through a surface or into a control volume. The moving pa ...
is the time derivative of
electric charge Electric charge is the physical property of matter that causes charged matter to experience a force when placed in an electromagnetic field. Electric charge can be ''positive'' or ''negative'' (commonly carried by protons and electrons respe ...
and so on. A common occurrence in physics is the time derivative of a
vector Vector most often refers to: *Euclidean vector, a quantity with a magnitude and a direction *Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism Vector may also refer to: Mathematic ...
, such as velocity or displacement. In dealing with such a derivative, both magnitude and orientation may depend upon time.


Example: circular motion

For example, consider a particle moving in a circular path. Its position is given by the displacement vector r=x\hat+y\hat, related to the angle, ''θ'', and radial distance, ''r'', as defined in the figure: :\begin x &= r \cos(\theta) \\ y &= r \sin(\theta) \end For this example, we assume that . Hence, the displacement (position) at any time ''t'' is given by :\mathbf(t) = r\cos(t)\hat+r\sin(t)\hat 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 :, \mathbf(t), = \sqrt=\sqrt = r\, \sqrt = r using the
trigonometric identity In trigonometry, trigonometric identities are Equality (mathematics), equalities that involve trigonometric functions and are true for every value of the occurring Variable (mathematics), variables for which both sides of the equality are defined. ...
and where \cdot 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: : \begin \mathbf(t) = \frac &= r \left frac, \frac \right\\ &= r\ -\sin(t),\ \cos(t)\\ &= y (t), x(t) \end 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 In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an algebra ...
: :\mathbf \cdot \mathbf = y, x\cdot
, y The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline ...
= -yx + xy = 0\, . Acceleration is then the time-derivative of velocity: :\mathbf(t) = \frac = x(t), -y(t)= -\mathbf(t)\, . 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 mechanics, acceleration is the rate of change of the velocity of an object with respect to time. Accelerations are vector quantities (in that they have magnitude and direction). The orientation of an object's acceleration is given by th ...
.


In differential geometry

In
differential geometry Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multili ...
, quantities are often expressed with respect to the local covariant basis, \mathbf_i , where ''i'' ranges over the number of dimensions. The components of a vector \mathbf expressed this way transform as a contravariant
tensor In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tensor ...
, as shown in the expression \mathbf=U^i\mathbf_i , invoking Einstein summation convention. If we want to calculate the time derivatives of these components along a trajectory, so that we have \mathbf(t)=U^i(t)\mathbf_i(t) , we can define a new operator, the invariant derivative \delta , which will continue to return contravariant tensors: :\begin \frac = \frac + V^j\Gamma^i_ U^k \\ \end where V^j=\frac (with x^j being the ''j''th coordinate) captures the components of the velocity in the local covariant basis, and \Gamma^i_ are the
Christoffel symbols In mathematics and physics, the Christoffel symbols are an array of numbers describing a metric connection. The metric connection is a specialization of the affine connection to surfaces or other manifolds endowed with a metric, allowing distanc ...
for the coordinate system. Note that explicit dependence on ''t'' has been repressed in the notation. We can then write: :\begin \frac = \frac \mathbf_i \\ \end as well as: :\begin \frac = \frac \mathbf_i \\ \end In terms of the covariant derivative, \nabla_, we have: :\begin \frac = V^j \nabla_ U^i \\ \end


Use in economics

In
economics Economics () is the social science that studies the Production (economics), production, distribution (economics), distribution, and Consumption (economics), consumption of goods and services. Economics focuses on the behaviour and intera ...
, many theoretical models of the evolution of various economic variables are constructed in
continuous time In mathematical dynamics, discrete time and continuous time are two alternative frameworks within which variables that evolve over time are modeled. Discrete time Discrete time views values of variables as occurring at distinct, separate "po ...
and therefore employ time derivatives.See for example One situation involves a stock variable and its time derivative, a flow variable. Examples include: * The flow of net
fixed investment Fixed investment in economics is the purchasing of newly produced fixed capital. It is measured as a flow variable – that is, as an amount per unit of time. Thus, fixed investment is the accumulation of physical assets such as machinery, land ...
is the time derivative of the
capital stock A corporation's share capital, commonly referred to as capital stock in the United States, is the portion of a corporation's equity that has been derived by the issue of shares in the corporation to a shareholder, usually for cash. "Share capi ...
. * The flow of
inventory investment Inventory investment is a component of gross domestic product (GDP). What is produced in a certain country is naturally also sold eventually, but some of the goods produced in a given year may be sold in a later year rather than in the year they wer ...
is the time derivative of the stock of
inventories Inventory (American English) or stock (British English) refers to the goods and materials that a business holds for the ultimate goal of resale, production or utilisation. Inventory management is a discipline primarily about specifying the sha ...
. * The growth rate of the
money supply In macroeconomics, the money supply (or money stock) refers to the total volume of currency held by the public at a particular point in time. There are several ways to define "money", but standard measures usually include Circulation (curren ...
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 Output may refer to: * The information produced by a computer, see Input/output * An output state of a system, see state (computer science) * Output (economics), the amount of goods and services produced ** Gross output in economics, the value o ...
is the time derivative of the flow of output divided by output itself. * The growth rate of the
labor force The workforce or labour force is a concept referring to the pool of human beings either in employment or in unemployment. It is generally used to describe those working for a single company or industry, but can also apply to a geographic reg ...
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 An interest rate is the amount of interest due per period, as a proportion of the amount lent, deposited, or borrowed (called the principal sum). The total interest on an amount lent or borrowed depends on the principal sum, the interest rate, th ...
can appear. * The
inflation rate In economics, inflation is an increase in the general price level of goods and services in an economy. When the general price level rises, each unit of currency buys fewer goods and services; consequently, inflation corresponds to a reductio ...
is the growth rate of the
price level The general price level is a hypothetical measure of overall prices for some set of goods and services (the consumer basket), in an economy or monetary union during a given interval (generally one day), normalized relative to some base set ...
—that is, the time derivative of the price level divided by the price level itself.


See also

* Differential calculus *
Notation for differentiation In differential calculus, there is no single uniform notation for differentiation. Instead, various notations for the derivative of a function or variable have been proposed by various mathematicians. The usefulness of each notation varies with ...
*
Circular motion 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 ro ...
* Centripetal force *
Spatial derivative {{unreferenced, date=July 2016 A spatial gradient is a gradient whose components are spatial partial derivatives, derivatives, i.e., rate of change (mathematics), rate of change of a given scalar (physics), scalar physical quantity with respect to ...
*
Temporal rate In mathematics, a rate is the ratio between two related quantities in different units. If the denominator of the ratio is expressed as a single unit of one of these quantities, and if it is assumed that this quantity can be changed systematicall ...


References

{{DEFAULTSORT:Time Derivative Differential calculus