

GRAD, DIV, CURL Origins
Name: David
Status: educator
Age: N/A
Location: N/A
Country: N/A
Date: 6/22/2005
Question:
In Vector Calculus why are GRAD, DIV and CURL called
GRAD, DIV and CURL?
Replies:
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
David,
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 nonzero 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 righthand 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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Update: June 2012

