For example, "gradient", in geography, the greater the height gradient, the steeper the slope; Gradient direction means that if you walk along the gradient direction, you will get the fastest downhill speed. The greater the temperature gradient, the more drastic the change of hot and cold, and so on.

There are also concepts such as "curl" and "divergence", and the corresponding practical phenomena can be found in physics (especially fluid mechanics)

If you are asking about the function in pure mathematics, it is to reflect how drastic the change of that quantity is; In multivariate calculus, which direction changes the most dramatically?