Vector to Multi-Index Partial Derivative

Operation

For a given vector $v=(v_1,v_2,\dots )$ For multi-index $(x_1,x_2,\dots )$ $\frac{\partial }{\partial v}=[\frac{\partial }{\partial v_1}{]}^{x_1}[\frac{\partial }{\partial v_2}{]}^{x_2}\dots$

Example

For vector $v=(x,y,z)$ For multi-index $(1,0,2)$ $\frac{\partial }{\partial v}={\left[\frac{\partial }{\partial x}\right]}^1{\left[\frac{\partial }{\partial y}\right]}^0{\left[\frac{\partial }{\partial z}\right]}^2=\frac{{\partial }^3}{\partial x\partial z^2}$