We have two independent variables, x and y, and one dependent variable f(x,y). The set of all points (x,y,z) with z=f(x,y) forms a surface in three-dimensional space, a special surface satisfying the vertical line test---no vertical line meets the surface in more than one point.
We will demonstrate this here with two examples. The first example will mainly be used for geometrical observations. The second example is the Cassini surface, yielding Cassini curves.
For every such function (surface) we get a family of curves in the plane, the level lines. For a fixed number (height) h, the level line of height h are all points (x,y) in the plane where f(x,y)=h.
fx and fy are again functions R2 ---> R, the partial derivatives of f.
The gradient vector is (fx,fy). It contains both direction and absolute value. The direction points into the direction of the steepest ascend. This direction is always perpendicular to the level line through this point (provided f is differentiable there). The absolute value, the square root of fx2+ fy2 tells us the slope of this steepest ascend.