Simplify the following expression involving the partial derivative with respect to variable u.

Question

Simplify the following expression $\frac{1}{||\partial_u \vec{X}||}\partial_u(\frac{\partial_u \vec{X}}{||\partial_u \vec X||})$

My attempt:- Let $\vec{X}=(X_1,X_2,...,X_n)$

$\frac{1}{||\partial_u \vec{X}||}\partial_u(\frac{\partial_u \vec{X}}{||\partial_u \vec X||})=\frac{1}{||\partial_u \vec{X}||}\partial_u(\frac{\partial_u X_1}{||\partial_u \vec{X}||},\frac{\partial_u X_2}{||\partial_u \vec{X}||},...,\frac{\partial_u X_n}{||\partial_u \vec{X}||})$

$=\frac{1}{||\partial_u \vec{X}||}(...,\partial_u(\frac{\partial_u X_n}{||\partial_u \vec{X}||)}))$

$=\frac{1}{||\partial_u \vec{X}||}(...,\frac{||\partial_u \vec{X}||\partial_u(\partial_u {X_n})-\partial_u {X_n}\partial(||\partial_u \vec{X}||)}{||\partial_u \vec{X}||^2})$ ($\because $ by quotient rule of partial differentiation)

$=\frac{1}{||\partial_u \vec{X}||}(...,\frac{||\partial_u \vec{X}||\partial_u(\partial_u {X_n})-\partial_u {X_n}\partial(||\partial_u \vec{X}||)}{||\partial_u \vec{X}||^2})$

$=\frac{1}{||\partial_u \vec{X}||}(...,\frac{||\partial_u \vec{X}||\partial_u(\partial_u {X_n})-\partial_u {X_n}\partial_u(||\partial_u \vec{X}||)}{||\partial_u \vec{X}||^2})$

$=\frac{1}{||\partial_u \vec{X}||}(...,\frac{||\partial_u \vec{X}||\partial_u(\partial_u {X_n})-\partial_u {X_n}(\frac{\vec{w}^T\partial_u \vec{w}}{||\vec{w}||})}{||\partial_u \vec{X}||^2})$ ($\because$ Result, Let $\vec{w}=\partial_u \vec{X})$

$=\frac{1}{||\partial_u \vec{X}||}(...,\frac{||\partial_u \vec{X}||\partial_u(\partial_u {X_n})-\partial_u {X_n}(\frac{\partial_u \vec{X}^T}{||\partial_u \vec{X}||})\partial_u \partial_u \vec{X}}{||\partial_u \vec{X}||^2})$

$=\frac{1}{||\partial_u \vec{X}||}(...,\frac{||\partial_u \vec{X}||(\partial_u^2 {X_n})-\partial_u {X_n}(\frac{\partial_u \vec{X}^T\partial_u^2 \vec{X}}{||\partial_u \vec{X}||})}{||\partial_u \vec{X}||^2})$

My questions: Is my calculations correct? Am I able to reduce further? Please help me.

Answer 1

Let's from the beginning write $w = \partial_uX$. Then we first compute $$\partial_j ||w|| = \partial_j \sqrt{w_1^2+..+w_n^2} = \frac{1}{2\sqrt{w_1^2+..+w_n^2}}\sum_{i=1}^n2w_i\partial_jw_i = \frac{w^T\partial_jw}{||w||}$$ Thus for the $i$-th component $w_i$ $$\partial_j\frac{w_i}{||w||} = \frac{||w||\partial_jw_i-w_i\partial_j||w||}{||w||^2}$$ by the quotient rule. Now substituting our result for $\partial_j||w||$ we get $$\partial_j\frac{w_i}{||w||} = \frac{||w||\partial_jw_i-w_i\frac{w^T\partial_jw}{||w||}}{||w||^2} = \frac{1}{||w||}\left(\partial_jw_i-\frac{w_i}{||w||^2}w^T\partial_jw\right)$$ And thus for the whole vector $$\partial_j\frac{w}{||w||} = \frac{1}{||w||}\left(\partial_jw-\frac{w}{||w||^2}w^T\partial_jw\right)$$ Hope that helps :)