Directional Derivative
Derivative along normal #
terms #
- ϕ=0: surface
- $\hat n$: normal vector at some point P
- dn: small change in position along $\hat n$
- r: general position coord <dx,dy,dz>
- dr: small change in generalized position coord
change in dr and dn in r causes the same change of δϕ in ϕ
derivation #
$$ \begin{align*} \frac{\partial \phi}{\partial n}=\lim _{\delta n \rightarrow 0} \frac{\delta \phi}{\delta n}\\ \lim _{\delta n \rightarrow 0} \frac{\nabla \phi \cdot \vec{dr}}{\delta n}\\ \lim _{\delta n \rightarrow 0} \frac{|\nabla \phi| \hat n \cdot \vec{dr}}{\delta n} \end{align*} $$
$\hat n \cdot \vec{dr}$: amount of dr pointing towards the normal vector = δn
$$ \begin{align*} \lim _{\delta n \rightarrow 0} \frac{|\nabla \phi| \delta n}{\delta n} &= |\nabla \phi | \end{align*} $$
Conclusion: $\text{grad }\vec \phi$ is a vector $\perp$ to surface and has a magnitude = rate of change of ϕ along normal.
Directional Derivative #
$$ \frac{\partial \phi}{\partial r}=\vec\nabla \phi\cdot \hat r $$
∴ the direction of steepest ascent of a scalar field ϕ is $\vec\nabla \phi$