Vector Analysis PDF

VECTOR ANALYSIS Notation for a vector: A SCALAR FIELD Scalar field means we take some space, say a plane, and measure some scalar value at each point. Say we have a big flat pan of shallow water sitting on the stove. If the water is shallow enough we can pretend that it is two-dimensi...

VECTOR ANALYSIS Notation for a vector: A SCALAR FIELD Scalar field means we take some space, say a plane, and measure some scalar value at each point. Say we have a big flat pan of shallow water sitting on the stove. If the water is shallow enough we can pretend that it is two-dimensional. Each point in the water has a temperature; the water over the stove flame is hotter than the water at the edges. But temperatures have no direction. There's no such thing as a north or an east temperature. The temperature is a scalar field: for each point in the water there is a temperature, which is a scalar, which says how hot the water is at that point. T(x,y,z) in 3D VECTOR FIELD A vector field means we take some space, say a plane, and measure some vector value at each point. Take the pan of water off the stove and give it a stir. Some of the water is moving fast, some slow, but this does not tell the whole story, because some of the water is moving north, some is moving east, some is moving northeast or other directions. Movement north and movement west could have the same speed, but the movement is not the same, because it is in different directions. To understand the water flow you need to know the speed at each point, but also the direction that the water at that point is moving. Speed in a direction is called a "velocity", and the velocity of the swirling water at each point is an example of a vector field. v ( x, y , z )  GRADIENT OF A SCALAR FIELD  ˆ  ˆ  ˆ  = i+ j+ k x y z     d = dx + dy + dz x y z  ˆ  ˆ  ˆ d = ( i + j+ k ).(dxiˆ + dyjˆ + dzkˆ) x y z d = ( ) (dr ) d =  dr cos   d    =   dr  max  PHYSICAL SIGNIFICANCE OF GRADIENT Is the the vector in the direction of most rapid  change of  and its magnitude is equal to the rate of change. Let us suppose H(x,y,z) is a scalar field indicating the height above sea level PROJECTION OF VECTOR A A  aˆ â â