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VECTOR CALCULUS

INTRODUCTION

Application of vector in flow of water

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.