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Del Operator
A calculus function called the del operator is fundamental to determining gradient, divergence and curl. It finds the change in flow for any point in space along the x, y, and z coordinates that define the familiar three dimensions. Performing the del operation on a single point can involve dozens of mathematical steps. Working them by hand for large groups of data would be an enormous task, though computers can do this rapidly, making innovations such as accurate weather forecasting possible. Mathematicians draw the del symbol as a small equilateral triangle pointing downward.
Gradient
Quantities such as weight and temperature consist of single numbers, like 15 degrees or 1,000 pounds. Scientists call these quantities scalars. Measurements like velocity and force, on the other hand, are vectors, having two numbers -- an amount and a direction. For example, the weatherman says the wind is out of the east at seven miles per hour. Scientists indicate vectors with arrows, as arrows have a length, indicating the magnitude or strength of the measurement, and point in a specific direction. The gradient is a vector resulting from a del operation on a surface. If the surface is flat, the gradient is zero; its shape doesn't change. If the surface is bumpy and hilly, the gradient points away from it. Where a surface has dips and valleys, the gradient points down into the surface. The more severe the bump or valley, the greater the gradient is at that point.
Divergence
Unlike the gradient, which is a vector, divergence is a simple number. It answers the questions, "Is something flowing into or out of this point?" and "How much?" Using divergence to analyze a bathtub with the faucet turned on and the stopper removed, most of the points in that space have a divergence of zero: Water neither flows out nor in. However, if you look at the area under the faucet, divergence becomes large. All the water flows into the tub from that point. Examining the drain, divergence is negative there, and also a large number, as all the water flows out of the tub at that point.
Curl
Curl is yet another way of looking at flows, and also comes from the del operator. Like the gradient measurement, curl is a vector. Looking at the bathtub example, as water drains out, it makes a little vortex or tornado-shaped funnel going into the drain. The curl of the flow is the intensity and direction of the vortex. If it spirals in a clockwise direction, the curl points down the drain; otherwise, it points out of the drain. If you look at all the other points in the bathtub, curl is zero, since the water spirals only at the drain.