Differentiability

We have introduced the limit and continuity of multivariate functions in previous section, then we come to the differentiability.

In single-variate functions, the differentiability and the derivative are the same thing, that is, using a tangent line to linearly approaching a function at somewhere. But in multivariate functions, they are different. Since the derivative is the way of using lines to approach functions, and the change of a surface in three space can not be represented by a line, or in one direction, so, intuitively and naturally, we try to use a plane to approach a surface in three space. That comes to the first concept in this section: local linear.

Local Linearity

The core idea of this concept is still linear approximation. We say:

If a function is local linearly, it can be written in the form of the sum of some linear items, and the error is convergent there.

Theoretically, we need to find the directional derivative in all directions to prove that a function is differentiable at a point, but by applying the method of linear composition, we only need to prove that this function can be approached in two different directions.

Luckily, we have a property that represent the linear approximation in a given direction which we have introduced in the previous section, that is, the partial derivative.

So, to prove that a function is differentiable at a given point, all we need to do is simply prove that all the partial derivatives are existed and continuous.
Note that this theorem CAN NOT be used conversely.

Gradient

Meanwhile, we can now represent a surface by a plane, with the linear composition of its partial derivatives as its tangent direction.

That is the concept of the gradient.

And an important relationship is revealed by the definition of the gradient, that is the gradient of a function must exists while the function is differentiable.

Directional Derivative

The directional derivative is actually the linear composition of partial derivatives in a given direction.