Hi gentlemen, Can anyone give a conceptual and geometric interpretation of vector fields,curl & divergence and also relate them to real world applications. thanks.
There is a book called "Div, Grad, Curl and All That" that has been around for decades. It is pretty well liked. The best description of this I've ever seen was in the introductory chapter of Electromagnetic Fields by Wangsness. Excellent coverage of vector calc.
Well to know what vector field is implies you know what a field is, so let us start there. In Physics a field is a point function which means that for some region in space we can assign a particular finite quantity to every point in that region. This quantity may be zero but not infinity. A region is section of space defined by coordinates x,y,z. For a scalar field we can assign a scalar value to every point so for instance in a bottle of fluid, every point in that bottle (the region) has a density and a pressure. These examples are easy to see and tangible. Less tangible are examples from electromagnetism, where every point in an electric field has an energy value or electric potential. For a vector field we assign a vector to every point in the region. So in a river we can assign a flow velocity vector to every point in the river. Remembering that vectors have both magnitude and direction. In electromagnetics we can assign a force vector acting on a unit charge located at every point in the region. This is normally what we mean by the 'electric field' in Physics, which is a vector field. So we have a spatial array of vectors (or scalars) that change from point to point. So we can do some calculus on this array of data, but since we are in at least two dimensions, we have to use partial differentiation, since the change can vary with direction. Grad, div and curl are the vector equivalents of this process. Do you remember your calculus101 definition of differentiation?