Define Del factor and explain.
Answers
In the Cartesian coordinate system Rn with coordinates {\displaystyle (x_{1},\dots ,x_{n})} (x_{1},\dots ,x_{n}) and standard basis {\displaystyle \{{\vec {e}}_{1},\dots ,{\vec {e}}_{n}\}} \{{\vec e}_{1},\dots ,{\vec e}_{n}\}, del is defined in terms of partial derivative operators as
{\displaystyle \nabla =\sum _{i=1}^{n}{\vec {e}}_{i}{\partial \over \partial x_{i}}=\left({\partial \over \partial x_{1}},\ldots ,{\partial \over \partial x_{n}}\right)} {\displaystyle \nabla =\sum _{i=1}^{n}{\vec {e}}_{i}{\partial \over \partial x_{i}}=\left({\partial \over \partial x_{1}},\ldots ,{\partial \over \partial x_{n}}\right)}
In three-dimensional Cartesian coordinate system R3 with coordinates {\displaystyle (x,y,z)} (x,y,z) and standard basis or unit vectors of axes {\displaystyle \{{\vec {e}}_{x},{\vec {e}}_{y},{\vec {e}}_{z}\}} \{{\vec e}_{x},{\vec e}_{y},{\vec e}_{z}\}, del is written as
{\displaystyle \nabla ={\vec {e}}_{x}{\partial \over \partial x}+{\vec {e}}_{y}{\partial \over \partial y}+{\vec {e}}_{z}{\partial \over \partial z}=\left({\partial \over \partial x},{\partial \over \partial y},{\partial \over \partial z}\right)} {\displaystyle \nabla ={\vec {e}}_{x}{\partial \over \partial x}+{\vec {e}}_{y}{\partial \over \partial y}+{\vec {e}}_{z}{\partial \over \partial z}=\left({\partial \over \partial x},{\partial \over \partial y},{\partial \over \partial z}\right)}
Del can also be expressed in other coordinate systems, see for example del in cylindrical and spherical coordinates.