Vector Calculus (11) - Differentiation

27. Differentiation of Scalar Function

 

Scalar Function $ \phi =\phi (s) $

in the $ x_{i},x_{i}' $ coordinates, $ \phi =\phi ', s=s' \Rightarrow d\phi =d\phi ', ds=ds' $

$ \frac{d\phi }{ds}=\frac{d\phi '}{ds'}=(\frac{d\phi }{ds})' $

 

 

28. Differentiationn of Vector Function

 

Vector Function $ \vec{A} $ w.r.t. a scalar S

$ A_{i}'=\sum_{j}^{}\lambda _{ij}A_{j} $

$ \frac{dA_{i}'}{ds'}=\frac{d}{ds'}\sum_{j}^{}\lambda _{ij}A_{j}=\sum_{j}^{}\lambda _{ij}\frac{dA_{j}}{ds'}=\sum_{j}^{}\lambda _{ij}\frac{dA_{j}}{ds} (s=s') $

$ \frac{dA_{j}}{ds} $: components of a vector (=$ \frac{d\vec{A}}{ds} $)

 

$ \Rightarrow \frac{d\vec{A}}{ds}=\lim_{\Delta s\rightarrow 0}\frac{\Delta \vec{A}}{\Delta s}=\lim_{\Delta s\rightarrow 0}\frac{\vec{A}(s+\Delta s)-\vec{A}(s)}{\Delta s} $

 

$ \frac{d}{ds}(\vec{A}+\vec{B})=\frac{d\vec{A}}{ds}+\frac{d\vec{B}}{ds} $

$ \frac{d}{ds}(\vec{A}\cdot \vec{B})=\frac{d\vec{A}}{ds}\cdot\vec{B} +\vec{A}\cdot \frac{d\vec{B}}{ds} $

$ \frac{d}{ds}(\vec{A}\times  \vec{B})=\frac{d\vec{A}}{ds}\times\vec{B}+\vec{A}\times \frac{d\vec{B}}{ds} $

$ \frac{d}{ds}(\phi \vec{A})=\frac{d\phi }{ds}\vec{A}+\phi \frac{d\vec{A}}{ds} $