-
Gradient of $\phi = \nabla \phi = $ grad $\phi = \Big(\frac{\p \phi}{\p x}, \frac{\p \phi}{\p y}, \frac{\p \phi}{\p z}\Big) = \fcolorbox{white}{pink}{the direction where
$\phi$ increases most rapidly}$
- Directional derivative in the direction $\vec u$:
$ \nabla \phi \cdot \vec u$
Make sure $\vec u$ is a unit vector.
- Normal vector for the surface $\phi(x,y,z)=$ constant:
$ \vec n = \frac{\nabla \phi }{|\nabla \phi |} $
- If normal vector is $\vec n = (n_1, n_2, n_3)$, then
Tangent plane at $(x_0,y_0,z_0)$ is $\mm n_1(x-x_0) + n_2 (y-y_0)+ n_3(z-z_0)=0 $
Normal line at $(x_0,y_0,z_0)$ is $\mm \frac{x-x_0}{n_1} = \frac{y-y_0}{n_2}= \frac{z-z_0}{n_3} $