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.