PlanetPhysics/Vector Functions

A vector function of position in space is a function

${\mathbf {V} }(x,y,z)$

which associates with each point $x,y,z$ in space a definite vector. The function may be broken up into its three components

${\mathbf {V} }(x,y,z)=V_{1}(x,y,z){\mathbf {\hat {i}} }+V_{2}(x,y,z){\mathbf {\hat {j}} }+V_{3}(x,y,z){\mathbf {\hat {k}} }$

Examples of vector functions are very numerous in physics. Already the function $\nabla {V}$ has occurred. At each point of space $\nabla {V}$ has in general a definite vector value. In mechanics of rigid bodies the velocity of each point of the body is a vector function of the position of the point. Fluxes of heat, electricity, magnetic force, fluids, etc., are all vector functions of position in space.

The scalar operator ${\mathbf {a} }\cdot \nabla$ may be applied to a vector function ${\mathbf {V} }$ to yield another vector function.

Let

${\mathbf {V} }=V_{1}(x,y,z){\mathbf {\hat {i}} }+V_{2}(x,y,z){\mathbf {\hat {j}} }+V_{3}(x,y,z){\mathbf {\hat {k}} }$

and

${\mathbf {a} }=a_{1}{\mathbf {\hat {i}} }+a_{2}{\mathbf {\hat {j}} }+a_{3}{\mathbf {\hat {k}} }$

Then

${\mathbf {a} }\cdot \nabla =a_{1}{\frac {\partial }{\partial x}}+a_{2}{\frac {\partial }{\partial y}}+a_{3}{\frac {\partial }{\partial z}}$

$\left({\mathbf {a} }\cdot \nabla \right){\mathbf {V} }=\left({\mathbf {a} }\cdot \nabla \right)V_{1}{\mathbf {\hat {i}} }+\left({\mathbf {a} }\cdot \nabla \right)V_{2}{\mathbf {\hat {j}} }+\left({\mathbf {a} }\cdot \nabla \right)V_{3}{\mathbf {\hat {k}} }$

and

$\left({\mathbf {a} }\cdot \nabla \right){\mathbf {V} }=\left(a_{1}{\frac {\partial V_{1}}{\partial x}}+a_{2}{\frac {\partial V_{1}}{\partial y}}+a_{3}{\frac {\partial V_{1}}{\partial z}}\right){\mathbf {\hat {i}} }+\left(a_{1}{\frac {\partial V_{2}}{\partial x}}+a_{2}{\frac {\partial V_{2}}{\partial y}}+a_{3}{\frac {\partial V_{2}}{\partial z}}\right){\mathbf {\hat {j}} }+\left(a_{1}{\frac {\partial V_{3}}{\partial x}}+a_{2}{\frac {\partial V_{3}}{\partial y}}+a_{3}{\frac {\partial V_{3}}{\partial z}}\right){\mathbf {\hat {k}} }$

more to come..

This is from the public domain text by Gibbs.