Why derivatives appear in the form of gradients or divergence?

Question

There are many physics problems whose mathematical equations have the same form.

At these problems we always get an equation with a gradient. And the derivatives appear in the form of a gradient or a divergence.

What are the reasons benind that?

Answer 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

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

The term "gradient" is typically used for functions with several inputs and a single output (a scalar field). Yes, you can say a line has a gradient (its slope), but using "gradient" for single-variable functions is unnecessarily confusing for Physicists (my opinion!).

Answer 2

Because there are three indistinguishable spatial dimensions and the three partial derivatives need to enter into physics equations in a way that doesn’t distinguish changes in one direction from changes in the other directions.

Another way of saying this is that $\partial/\partial x$, $\partial/\partial y$, and $\partial/\partial x$ only make sense in isotropic 3D space when combined into the vector operator $\vec{\nabla}$, which is then used to form the gradient, the divergence, and the curl.