Directional Derivatives

The directional derivative of a function f at the point $\left(\vphantom{ a,b,c}\right.$a, b, c$\left.\vphantom{ a,b,c}\right)$ in the direction u = $\left(\vphantom{ u_{1},u_{2},u_{3}}\right.$u1, u2, u3$\left.\vphantom{ u_{1},u_{2},u_{3}}\right)$ is given by the inner product of ∇f and u at the point $\left(\vphantom{ a,b,c}\right.$a, b, c$\left.\vphantom{ a,b,c}\right)$. That is, for a vector u of unit length and a scalar function f,

Duf$\displaystyle \left(\vphantom{ a,b,c}\right.$a, b, c$\displaystyle \left.\vphantom{ a,b,c}\right)$ = f$\displaystyle \left(\vphantom{ a,b,c}\right.$a, b, c$\displaystyle \left.\vphantom{ a,b,c}\right)$u  
  = $\displaystyle {\frac{{\partial f}}{{\partial x}}}$$\displaystyle \left(\vphantom{ a,b,c}\right.$a, b, c$\displaystyle \left.\vphantom{ a,b,c}\right)$u1 + $\displaystyle {\frac{{\partial f}}{{%
\partial y}}}$$\displaystyle \left(\vphantom{ a,b,c}\right.$a, b, c$\displaystyle \left.\vphantom{ a,b,c}\right)$u2 + $\displaystyle {\frac{{\partial f}}{{\partial z}}}$$\displaystyle \left(\vphantom{ a,b,c}\right.$a, b, c$\displaystyle \left.\vphantom{ a,b,c}\right)$u3  


$\blacktriangleright$ To compute the directional derivative of f (x, y, z) = xyz in the direction

u = $\displaystyle \left(\vphantom{ \cos \frac{\pi }{8}\sin \frac{\pi }{9},\sin \frac{\pi }{8}%
\sin \frac{\pi }{9},\cos \frac{\pi }{9}}\right.$cos$\displaystyle {\frac{{\pi }}{{8}}}$sin$\displaystyle {\frac{{\pi }}{{9}}}$, sin$\displaystyle {\frac{{\pi }}{{8}%
}}$sin$\displaystyle {\frac{{\pi }}{{9}}}$, cos$\displaystyle {\frac{{\pi }}{{9}}}$$\displaystyle \left.\vphantom{ \cos \frac{\pi }{8}\sin \frac{\pi }{9},\sin \frac{\pi }{8}%
\sin \frac{\pi }{9},\cos \frac{\pi }{9}}\right)$

1.
Select xyz.

2.
While holding down the CTRL key, choose Evaluate.

3.
Choose Evaluate (or Evaluate Numerically).

$\blacktriangleright$ CTRL + Evaluate, Evaluate, Evaluate Numerically

xyz$\left(\vphantom{ \cos \frac{\pi }{8}\sin \frac{\pi }{9},\sin \frac{%
\pi }{8}\sin \frac{\pi }{9},\cos \frac{\pi }{9}}\right.$cos${\frac{{\pi }}{{8}}}$sin${\frac{{\pi }}{{9}}}$, sin${\frac{{%
\pi }}{{8}}}$sin${\frac{{\pi }}{{9}}}$, cos${\frac{{\pi }}{{9}}}$$\left.\vphantom{ \cos \frac{\pi }{8}\sin \frac{\pi }{9},\sin \frac{%
\pi }{8}\sin \frac{\pi }{9},\cos \frac{\pi }{9}}\right)$

     = $\left(\vphantom{ yz,xz,xy}\right.$yz, xz, xy$\left.\vphantom{ yz,xz,xy}\right)$$\left(\vphantom{ \cos \frac{\pi }{8}\sin \frac{%
\pi }{9},\sin \frac{\pi }{8}\sin \frac{\pi }{9},\cos \frac{\pi }{9}}\right.$cos${\frac{{\pi }}{{8}}}$sin${\frac{{%
\pi }}{{9}}}$, sin${\frac{{\pi }}{{8}}}$sin${\frac{{\pi }}{{9}}}$, cos${\frac{{\pi }}{{9}}}$$\left.\vphantom{ \cos \frac{\pi }{8}\sin \frac{%
\pi }{9},\sin \frac{\pi }{8}\sin \frac{\pi }{9},\cos \frac{\pi }{9}}\right)$

     = ${\frac{{1}}{{2}}}$yz$\sqrt{{\left( 2+\sqrt{2}\right) }}$sin${\frac{{1}}{{9}}}$π + ${\frac{{1}}{{2}}}$xz$\sqrt{{\left( 2-\sqrt{2}\right) }}$sin${\frac{{1}}{{9}}}$π + xy cos${\frac{{1}}{{9}}}$π

     = .31599yz + .13089xz + .93969xy