I know that a simple vector-valued function can map out a curve (for example, r(t) = <cos(t), sin(t), 1> is a circle). As well, I know that a simple multivariable function can map out a surface (for example, f(x,y) = x^2 + y^2 is an elliptical paraboloid). But I can't seem to wrap my head around what a multivariable vector-valued function is, like f(x,y) = xi^ + yj^ (pardon my notation, I just mean x multiplied by the i-hat unit vector and y multiplied by the j-hat unit vector). Can this function be thought of in any sort of physical representation that makes it more understandable?
I've heard the term vector field being thrown around in the context of these kinds of functions, and I understand what they are, but I don't get how I would visualize (for example) f(x,y) = 3xi^ + 0.5yj^ || or || f(x,y) = (x^2)i^ + (sqrt(y))j^ as vector fields...