Relationship between directional derivative and gradient in x, y and z

Question

1. Can anybody explain the relationship between directional derivative and gradient?

2. What can I use the results of directional derivative and gradient for ?

Answer 1

The directional derivative is defined as

$$f'(x;v)=\lim_{t>0,\,t\to 0} \frac{f(x+tv)-f(x)}{t}.$$

It can exist even if the function $f$ is not differentiable. If the directional derivative is actually linear in the direction $v$, $f'(x;v)=Av$, one can assign this linear function $A$ as the total derivative. However, $f$ is only (Frechet) differentiable in $x$ if

$$f(x+v)=f(x)+Av+o(\|v\|)$$

that is, if the approximation by the linearization is uniformly good over all directions. Then $A$ is also written as $f'(x)$ or $Df_x$ or similar.

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Now where does the gradient enter? Of course with a scalar product. The derivative $f'(x)$ of a scalar function, i.e., its Jacobian, is covector, linear functional, or in cartesian space, a row vector. The gradient $∇f(x)$ is the vector (in cartesian space column vector) associated to that row vector via

$$\langle \nabla f(x),v\rangle=f'(x)v.$$

In the standard euclidean setting in cartesian space, this relation is simple transposition, $$\nabla f(x)=f'(x)^T,$$ in curved space with non-trivial metric tensors, the relation is more complicated.