A time derivative is a
derivative
In mathematics, the derivative is a fundamental tool that quantifies the sensitivity to change of a function's output with respect to its input. The derivative of a function of a single variable at a chosen input value, when it exists, is t ...
of a function with respect to
time
Time is the continuous progression of existence that occurs in an apparently irreversible process, irreversible succession from the past, through the present, and into the future. It is a component quantity of various measurements used to sequ ...
, usually interpreted as the rate of change of the value of the function. The variable denoting time is usually written as
.
Notation
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
In calculus, the second derivative, or the second-order derivative, of a function is the derivative of the derivative of . Informally, the second derivative can be phrased as "the rate of change of the rate of change"; for example, the secon ...
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,
:
Use in physics
Time derivatives are a key concept in
physics
Physics is the scientific study of matter, its Elementary particle, 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 whi ...
. For example, for a changing
position , its time derivative
is its
velocity
Velocity is a measurement of speed in a certain direction of motion. It is a fundamental concept in kinematics, the branch of classical mechanics that describes the motion of physical objects. Velocity is a vector (geometry), vector Physical q ...
, and its second derivative with respect to time,
, is its
acceleration
In mechanics, acceleration is the Rate (mathematics), rate of change of the velocity of an object with respect to time. Acceleration is one of several components of kinematics, the study of motion. Accelerations are Euclidean vector, vector ...
. 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 cause an Physical object, object to change its velocity unless counterbalanced by other forces. In mechanics, force makes ideas like 'pushing' or 'pulling' mathematically precise. Because the Magnitu ...
is the time derivative of
momentum
In Newtonian mechanics, momentum (: momenta or momentums; 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. ...
*
power is the time derivative of
energy
Energy () is the physical quantity, quantitative physical property, property that is transferred to a physical body, body or to a physical system, recognizable in the performance of Work (thermodynamics), work and in the form of heat and l ...
*
electric current
An electric current is a flow of charged particles, such as electrons or ions, moving through an electrical conductor or space. It is defined as the net rate of flow of electric charge through a surface. The moving particles are called charge c ...
is the time derivative of
electric charge
Electric charge (symbol ''q'', sometimes ''Q'') is a physical property of matter that causes it to experience a force when placed in an electromagnetic field. Electric charge can be ''positive'' or ''negative''. Like charges repel each other and ...
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
* Disease vector, an agent that carries and transmits an infectious pathogen into another living organism
Vector may also refer to:
Mathematics a ...
, 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
, related to the angle, ''θ'', and radial distance, ''r'', as defined in the figure:
:
For this example, we assume that . 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 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
In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a Scalar (mathematics), scalar as a result". It is also used for other symmetric bilinear forms, for example in a pseudo-Euclidean space. N ...
:
:
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 mechanics, acceleration is the rate of change of the velocity of an object with respect to time. Acceleration is one of several components of kinematics, the study of motion. Accelerations are vector quantities (in that they have magn ...
.
In differential geometry
In
differential geometry
Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of Calculus, single variable calculus, vector calculus, lin ...
, 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
In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects associated with a vector space. Tensors may map between different objects such as vectors, scalars, and even other ...
, 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:
:
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
In mathematics and physics, covariance is a measure of how much two variables change together, and may refer to:
Statistics
* Covariance matrix, a matrix of covariances between a number of variables
* Covariance or cross-covariance between ...
,
, we have:
:
Use in economics
In
economics
Economics () is a behavioral science that studies the Production (economics), production, distribution (economics), distribution, and Consumption (economics), consumption of goods and services.
Economics focuses on the behaviour and interac ...
, many theoretical models of the evolution of various economic variables are constructed in
continuous time 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 is the time derivative of the
capital stock.
* The flow of
inventory investment is the time derivative of the stock of
inventories
Inventory (British English) or stock (American English) is a quantity of 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 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
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 valu ...
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
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, ...
can appear.
* The
inflation rate
In economics, inflation is an increase in the average price of goods and services in terms of money. This increase is measured using a price index, typically a consumer price index (CPI). When the general price level rises, each unit of curre ...
is the growth rate of the
price level—that is, the time derivative of the price level divided by the price level itself.
See also
*
Differential calculus
In mathematics, differential calculus is a subfield of calculus that studies the rates at which quantities change. It is one of the two traditional divisions of calculus, the other being integral calculus—the study of the area beneath a curve. ...
*
Notation for differentiation
In differential calculus, there is no single standard notation for differentiation. Instead, several notations for the derivative of a Function (mathematics), function or a dependent variable have been proposed by various mathematicians, includin ...
*
Circular motion
*
Centripetal force
Centripetal force (from Latin ''centrum'', "center" and ''petere'', "to seek") is the force that makes a body follow a curved trajectory, path. The direction of the centripetal force is always orthogonality, orthogonal to the motion of the bod ...
*
Spatial derivative
*
Temporal rate
References
{{DEFAULTSORT:Time Derivative
Differential calculus