Directional derivative

In multivariable calculus, there is  a notion of derivative, that is directional derivative.

Directional derivative is special derivative, follows a vector.

Definition. Let U \subset \mathbb{R}^m and U is open set. Let f: U \to \mathbb{R}^n. Suppose a \in U. Given v \in \mathbb{R}^m with v \ne 0, the directional of f at a corresponding to v is:

Df_v(a) = \lim\limits_{t \to 0}\dfrac{f(a + tu) - f(a)}{t}, if the limit exists.

Another definition: The directional derivative of f at a \in U corresponding to v is Df(a)\cdot v.

Theorem. Two above definitions are equivalent.



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