Gradient,divergence and curl.

1. It will tell you about the angular velocity/accleration at that point, but that's not very accurate.

2. If we jump ahead to 5 (which I'll explain later) and look at its definition, we can now define the components of the curl field along the xyz axes. Since the shape of the infinitesemal loop is not important (assuming everything is smooth enough), I would suggest taking a rectangular with sides parallel to the different axes (while computing you will have three of this loops). Calculating the circulation density:

[tex]curl(F) _{x}=lim \frac{\oint_{C_{yz}}\vec{F}\vec{dr}}{A_{yz}}[/tex]
(C_{yz} is a rectangular loop parallel to the yz plain)

With this specific shape, will give you exactly the definitions of the derivatives. Summing all the components, you will recognize the structure of a cross product, or as you said, the determinant formulation.

A displacement vector is usually a function of t. With curl we are talking about functions of (x,y,z).

5. The projection of the curl against a specific direction, will tell you how much the field circulates around that direction at some point.

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!!