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A vector field is usually the source of the circulation.Curl is simply the
circulation per unit area, circulation density, or rate of rotation (amount of
twisting at a single point).Theflow of water will not usually be a constant in
space. It will be some function of position.
The flow of water is given in vector representation
Divergence and Curl
A vector field is a vector valued function of position in space. Let V = (u, v) be a vector field
representing the velocity of a fluid at a point (x, y) in two dimensions. Divergence ( ) is a
measure of how much a field comes together or flies apart where ∇=i⃗ ∂∂ x
+ j⃗ ∂∂ y
+ k⃗ ∂∂ z
Positive divergence means that the field is "flowing" out of a region.
Negative divergence means that the field is "flowing" into a region.
When the divergence is zero, then the amount that "flows" in must be equal to the amount
that flows out.The physical meanings of the divergence can be visualized as follows.
Curl is how much the field “twists”. For some idea of what curl does to a field; in a
whirlpool or a tornado all of the curl is in the center funnel, and the field (wind) wraps around
where the curl is.
Application of Gradient in Microwave oven
Example :The temperature inside the microwave oven is given by the temperature T = xyz . Find
the gradient of T at (3, 3,2).
Solution:
Application of Vectors in Sports
The outfielder in the baseball game moving a certain direction for a specific distanceto reach
a high fly ball before it touches the ground. The outfielder can’t just run directly for where he
sees the ball first or he is going to miss it by a long shot. The player must anticipate what
direction and how far the ball will be from him when it drops and move to that location to
have the best chance of catching the ball.
APPLICATION OF GREEN’S THEOREM
Example: Find the area of the region enclosed by the irregular cloth.
Application of Green’s theorem in planimeter
The planimeter is a mechanical device for measuring areas of regions in the plane which are
bounded by smooth boundaries. The measurement is based directly on Green's theorem in
multi-variable calculus: the planimeter integrates a line integral of a vector field which has
constant curl
.
Planimeters are mechanical instruments which can measure the area of closed
regions in the plane. Planimeters are used in medicine for example to measure the
size of the cross-sections of tumors or organs, in biology to measure the area of
leaves or wing sizes of insects, in agriculture to measure the area of forests, in
ingeneering it is used to measure the size of profiles.