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Name: David
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Date: 6/22/2005

In Vector Calculus why are GRAD, DIV and CURL called GRAD, DIV and CURL?

I think they are somewhat descriptive of what the operators do.

GRAD computes the gradient of a scalar function. That is, it finds the gradient, the slope, how fast you change, in any given direction.

DIV computes the divergence of a vector function. That is, it finds how much "stuff" is leaving a point in space.

CURL is a bit more obscure I suppose, at least as to why it is labeled such. It computes the rotational aspects of a vector function, maybe people thought how vectors "curl" around a center point, like wind curling around a low pressure on a weather map.

Steve Ross


Gradient, Divergence, and Curl are named based on what they actually measure.

A gradient is applied to a scalar quantity that is a function of a 3D vector field: position. The gradient measures the direction in which the scalar quantity changes the most, as well as the rate of change with respect to position. A common example of this is height as a function of latitude and longitude, often applied to mountain ranges. A measure of the slope, and direction of the slope, is often called the gradient.

A divergence is applied to a vector as a function of position, yielding a scalar. The divergence actually measures how much the vector function is spreading out. If you are at a location from which the vector field tends to point away in all directions, you will definitely have a positive divergence. If the field points inward toward a point, the divergence in and near that point is negative. If just as much of the vector field points in as out, the divergence will be approximately zero.

A curl measures just that, the curl of a vector field. Unlike the divergence, a curl yields a vector. A vector field that tends to point around an axis, such as vectors pointing tangential to a circle, will yield a non-zero curl with the axis around which the curl occurs as the direction. Another example is the velocity field of motion spiraling in or out, such as a whirlpool. Point your right-hand thumb along the direction of the curl. Curl your fingers around this axis. They will curl in the same direction as the vector field. I do not know the names of the texts, but I know there are books available with vector fields to illustrate both divergence and curl.

Dr. Ken Mellendorf
Physics Instructor
Illinois Central College

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