How do we define and calculate divergence in vector fields?

[Urmi Roy]

Please forgive me for extending even further...just my last two questions..

1. When we calculate curl,we reduce the area over which the path integral is being integrated to a single point--however,due to this, the magnitude of curl should always be very small,as the path integral over an area as small al that would be near to zero!
Am I saying something wrong?

2.Direction of curl is defined as the direction of maximum rotation...what doe the 'maximum' imply?

 

[Lord Crc]

1. When we calculate curl,we reduce the area over which the path integral is being integrated to a single point--however,due to this, the magnitude of curl should always be very small,as the path integral over an area as small al that would be near to zero!

But you divide by the (near-zero) area as well. This is similar to how the derivative is defined. The difference f(x+h)-f(x) will also be nearly zero (for continuous functions) as h goes to zero, but you divide by h, and so you get something else for the limit.