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 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 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 economics, many theoretical models of the evolution of various economic variables are constructed in continuous time and therefore employ time derivatives. 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