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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 $\vec\nabla \phi$