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Snap, [6] or jounce, [2] is the fourth derivative of the position vector with respect to time, or the rate of change of the jerk with respect to time. [4] Equivalently, it is the second derivative of acceleration or the third derivative of velocity, and is defined by any of the following equivalent expressions: = ȷ = = =.
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 higher order derivatives can be applied in physics; for example, while the first derivative of the position of a moving object with respect to time is the object's velocity, how the position changes as time advances, the second derivative is the object's acceleration, how the velocity changes as time advances.
This states that differentiation is the reverse process to integration. Differentiation has applications in nearly all quantitative disciplines. In physics, the derivative of the displacement of a moving body with respect to time is the velocity of the body, and the derivative of the velocity with respect to time is acceleration.
The last expression is the second derivative of position (x) with respect to time. On the graph of a function , the second derivative corresponds to the curvature or concavity of the graph. The graph of a function with a positive second derivative is upwardly concave, while the graph of a function with a negative second derivative curves in the ...
Taking a derivative with respect to a quantity divides the dimension by the dimension of the variable that is differentiated with respect to. Thus: position (x) has the dimension L (length); derivative of position with respect to time (dx/dt, velocity) has dimension T −1 L—length from position, time due to the gradient;
A natural example of a differential field is the field of rational functions in one variable over the complex numbers, (), where the derivation is differentiation with respect to . More generally, every differential equation may be viewed as an element of a differential algebra over the differential field generated by the (known) functions ...
In this article, we place a negative sign on the spatial coordinates (the time-positive metric convention = [,,,]). The factor of (1/ c ) is to keep the correct unit dimensionality , [length] −1 , for all components of the 4-vector and the (−1) is to keep the 4-gradient Lorentz covariant .
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