Consider a topographical map that gives the elevation of a region as a function of x and y. How can we find the highest point on a mountain or the lowest point in a valley? Are there theorems that provide criteria for a local maximum or minimum?
Let a set of equations describe the evolution of a tornado. At each location (three spatial coordinates), and at each time (another input variable), the equations specify the winds velocity in the x, y, and z directions, hence three output variables. Can we use calculus to describe how these quantities change with time?