Gradient

When we examined divergence, we came to understand that it lets us obtain a scalar field from a vector field — for example, obtain the distribution of charge from the vectors of an electric field's strength. The gradient, on the other hand, essentially does the opposite: it lets us obtain a vector field from a scalar field. Let us reason about this.

Everyone understands what a derivative is; for example, for a function $f(x)$ the derivative looks like $g(x)$ $= f^{\prime}(x)$, that is, it is also a function of the same argument. Its meaning is to show the rate of change of the function's value. In three-dimensional space a function depends on three arguments, so at a specific point the derivative must also depend on three arguments. That is all well and good — let us take the partial derivatives, but then what, add them up or multiply them?

What does a negative derivative tell us? The same thing that a negative acceleration tells us — the speed begins to fall, that is, it may well be greater than zero, but it is falling. So the derivative of a function of one variable shows whether the function is increasing or decreasing, that is, it shows the direction of change. This is important! In three-dimensional space we must also know the direction of change. So from the partial derivatives we form a vector, which is called the gradient.

where $(\bold{i}, \bold{j}, \bold{k})$ — the direction vectors of the coordinate system.

It is convenient to write the gradient as the nabla operator applied to the value of the scalar field at a point