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vector functions (Topic)

A vector function of position in space is a function

$\displaystyle {\bf V}(x, y, z)$

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

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

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 ${\bf a} \cdot \nabla$ may be applied to a vector function ${\bf V}$ to yield another vector function.

Let

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

and

$\displaystyle {\bf a} = a_1{\bf\hat{i}} + a_2 {\bf\hat{j}} + a_3{\bf\hat{k}} $

Then

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

$\displaystyle \left ( {\bf a} \cdot \nabla \right ) {\bf V} = \left ( {\bf a} \... ...right) V_2 {\bf\hat{j}} + \left ( {\bf a} \cdot \nabla \right) V_3 {\bf\hat{k}}$

and

$\displaystyle \left ( {\bf a} \cdot \nabla \right ) {\bf V} = \left ( a_1\frac{... ...al V_3}{\partial y} + a_3\frac{\partial V_3}{\partial z} \right ) {\bf\hat{k}} $

more to come..

This is from the public domain text by Gibbs.



"vector functions" is owned by bloftin.

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Cross-references: domain, operator, scalar, heat, fluxes, velocity, rigid bodies, mechanics, position, function, vector
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This is version 1 of vector functions, born on 2006-08-25.
Object id is 220, canonical name is VectorFunctions.
Accessed 1274 times total.

Classification:
Physics Classification: 02. (Mathematical methods in physics)
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