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The areal velocity magnitude (i.e., the areal speed) is this region's area divided by the time interval Δt in the limit that Δt becomes vanishingly small. The vector direction is postulated to be normal to the plane containing the position and velocity vectors of the particle, following a convention known as the right hand rule .
This article uses the standard notation ISO 80000-2, which supersedes ISO 31-11, for spherical coordinates (other sources may reverse the definitions of θ and φ): . The polar angle is denoted by [,]: it is the angle between the z-axis and the radial vector connecting the origin to the point in question.
Newton illustrates his formula with three examples. In the first two, the central force is a power law, F(r) = r n−3, so C(r) is proportional to r n. The formula above indicates that the angular motion is multiplied by a factor k = 1/ √ n, so that the apsidal angle α equals 180°/ √ n.
Since the speed v is likewise unchanging, the areal velocity 1 ⁄ 2 vr ⊥ is a constant of motion; the particle sweeps out equal areas in equal times. The area A of a circular sector equals 1 ⁄ 2 r 2 φ = 1 ⁄ 2 r 2 ωt = 1 ⁄ 2 r v φ t. Hence, the areal velocity dA/dt equals 1 ⁄ 2 r v φ = 1 ⁄ 2 h.
Terminal velocity depends on atmospheric drag, the coefficient of drag for the object, the (instantaneous) velocity of the object, and the area presented to the airflow. Apart from the last formula, these formulas also assume that g negligibly varies with height during the fall (that is, they assume constant acceleration).
For example, traveling a steady 50 mph for 3 hours results in a total distance of 150 miles. Plotting the velocity as a function of time yields a rectangle with a height equal to the velocity and a width equal to the time elapsed. Therefore, the product of velocity and time also calculates the rectangular area under the (constant) velocity curve.
Flux F through a surface, dS is the differential vector area element, n is the unit normal to the surface. Left: No flux passes in the surface, the maximum amount flows normal to the surface. Right: The reduction in flux passing through a surface can be visualized by reduction in F or dS equivalently (resolved into components, θ is angle to ...
The curl of a vector field F, denoted by curl F, or , or rot F, is an operator that maps C k functions in R 3 to C k−1 functions in R 3, and in particular, it maps continuously differentiable functions R 3 → R 3 to continuous functions R 3 → R 3.