The tangent plane to the graph of a function z=f(x,y) at a point (p,q) is the best linear function which approximates the function near (p,q).

The directional derivative is the slope in each direction of the tangent plane.

The gradient is a field of vectors which points to in the direction in which the function has its greatest growth.
z= x^2 +y^2 - 1
z= 1 - x^2 - y^2
z= x^2 - y^2

In order to find maximum and minimum of a function of several variable, look for the critical points, i.e., points where the gradient is the zero vector. In several variables it possible for the set of critical points to be quite large, for example z=(1-x^2-y^2)^2 has the unit circle plus the origin as its set of critical points. See if you can guess what the surface z=(1-sqrt(x^2-y^2))^2 looks like.

The second derivative test applied to z=xy(1-x-y) yields saddle points at (0,0),(0,1),(1,0) and a local max at (1/3,1/3). Can you see them?

The double integral is formed by adding up the volumes of boxes whose base gets smaller and smaller.