Time derivative
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 .
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
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,
Use in physics
Time derivatives are a key concept in
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
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 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
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
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
where (with being the jth 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:
Use in economics
In
- The flow of net capital stock.
- The flow 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 forceis 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.
See also
- Differential calculus
- Notation for differentiation
- Circular motion
- Centripetal force
- Spatial derivative
- Temporal rate
References
- ^ 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.
- ISBN 0-07-053667-8.