Review and deep understanding about "gradient"

First a very through post about gradient:
http://betterexplained.com/articles/vector-calculus-understanding-the-gradient/

(1) The gradient is a fancy word for derivative, or the rate of change of a function. It’s a vector (a direction to move) that

• Points in the direction of greatest increase of a function (intuition on why)
• Is zero at a local maximum or local minimum (because there is no single direction of increase

(2) The key insight is to recognize the gradient as the generalization of the derivative. The gradient points to the maximum of the function; follow the gradient, and you will reach the local maximum.