PlanetPhysics/Vector Functions

A [[../Vectors/|vector]] [[../Bijective/|function]] of [[../Position/|position]] in space is a function

$𝐕(x,y,z)$

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

$𝐕(x,y,z)=V_1(x,y,z)\widehat{𝐢}+V_2(x,y,z)\widehat{𝐣}+V_3(x,y,z)\widehat{𝐤}$

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

The [[../Vectors/|scalar]] [[../QuantumSpinNetworkFunctor2/|operator]] $𝐚\cdot \mathrm{\nabla }$ may be applied to a vector function $𝐕$ to yield another vector function.

Let

$𝐕=V_1(x,y,z)\widehat{𝐢}+V_2(x,y,z)\widehat{𝐣}+V_3(x,y,z)\widehat{𝐤}$

and

$𝐚=a_1\widehat{𝐢}+a_2\widehat{𝐣}+a_3\widehat{𝐤}$

Then

$𝐚\cdot \mathrm{\nabla }=a_1\frac{\partial }{\partial x}+a_2\frac{\partial }{\partial y}+a_3\frac{\partial }{\partial z}$

$(𝐚\cdot \mathrm{\nabla })𝐕=(𝐚\cdot \mathrm{\nabla })V_1\widehat{𝐢}+(𝐚\cdot \mathrm{\nabla })V_2\widehat{𝐣}+(𝐚\cdot \mathrm{\nabla })V_3\widehat{𝐤}$

and

$(𝐚\cdot \mathrm{\nabla })𝐕=(a_1\frac{\partial V_1}{\partial x}+a_2\frac{\partial V_1}{\partial y}+a_3\frac{\partial V_1}{\partial z})\widehat{𝐢}+(a_1\frac{\partial V_2}{\partial x}+a_2\frac{\partial V_2}{\partial y}+a_3\frac{\partial V_2}{\partial z})\widehat{𝐣}+(a_1\frac{\partial V_3}{\partial x}+a_2\frac{\partial V_3}{\partial y}+a_3\frac{\partial V_3}{\partial z})\widehat{𝐤}$

more to come..

This is from the public [[../Bijective/|domain]] text by Gibbs.