In which direction should the directional derivative be maximal? I know that it's the gradient of $f$ but in some textbook it was mentioned that it's the unit vector of $\nabla f$. Why?

  • 3
    $\begingroup$ The direction of a non-zero vector, and the unit vector you obtain from it by dividing it by its length, are surely the same? $\endgroup$ – Lord Shark the Unknown Aug 5 '17 at 8:39
  • $\begingroup$ The main reason to define direction by unit vectors is that when you need to compute projections of other vectors to that direction, you only need to compute the dot product: If $|v|=1$, the projection of $x$ in the direction of $v$ is $(x\cdot v)v$. $\endgroup$ – Marja Aug 5 '17 at 9:46

In terms of direction in $\mathbf{R}^n$, any two vectors which are positive scalar multiples of each other denote the same direction. So, $\nabla f$ gives the same direction as $\lambda \nabla f$ for $\lambda\in \mathbf{R}_{>0}$. In particular, $\frac{\nabla f}{\lvert \nabla f\rvert}$ gives us the same direction as $\nabla f$.


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