I'm trying to get a better intuition for scalar and vector fields, so thought I'd experiment with Plot3D .. For example I wanted to plot a function like x^2 + xy + y^2, easy enough. Then I wanted to see what the gradient looked like - could only find an abstruse set of nested commands. I will also want to look at divergence and curl, so combinations of scalars and vectors plus these operators .... Is there a simple and easy to remember way to do things like this? Thanks.

the Wolfram Demonstrations Project has a number of useful bits for gradient, but nothing along those lines for divergence and curl - so maybe someone will see this and build some demonstrations for those as well :-) $X^2 + x*y + y^2$ is just an example off the top of my head, another I liked was at https://www.wolfram.com/mathematica/new-in-8/new-and-improved-scientific-and-information-visualization/show-the-gradient-field-on-a-surface.html and of course some trig combination functions ....anyway, thanks for your interest & time David

You can find some examples and pictures of these items in Wikipedia or perhaps other places on the web. You could look up Grad, Div, Curl, VectorPlot, VectorPlot3D, ContourPlot and ContourPlot3D in Mathematica Documentation but you will probably not be able to put them together in a short time. You might look in the Wolfram Demonstrations Project for examples. Unfortunately, Mathematica is not quite an off-the-self and start using application and I don't know of an existing direct application that would help you.

Thanks David, while you make a great deal of sense, I am writing an exam in the not-too-distant and for now my goal is just to plot a few simple 3-D functions with their associated grad/div/curl - to help me visualise them and the differences between them; I don't know mathematica at all well and don't want to get side-tracked (now) in trying to learn it well enough. My thought was that I could find examples of each of any simple function with it's grad/div/curl and could code them in a macro - I could get a lot of visualising done quite time-cheaply. Do you (or anyone) perhaps know of a simple example of 'recording a macro' I could look at?

Simple way to plot functions & vector properties