vector functions | Physics Library

A vector function of 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) = V1(x, y,z)ˆi + V2(x, y,z)ˆj + V3 (x, y,z)ˆk

Examples of vector functions are very numerous in physics. Already the function ∇V has occurred. At each point of space ∇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 a ⋅∇ may be applied to a vector function V to yield another vector function.

Let

V = V (x,y,z)ˆi + V (x,y,z)ˆj + V (x,y,z)ˆk 1 2 3

and

ˆ ˆ ˆ a = a1 i + a2j + a3k

Then

∂ ∂ ∂ a ⋅ ∇ = a1 ---+ a2--- + a3--- ∂x ∂y ∂z

(a ⋅ ∇) V = (a ⋅ ∇ ) V ˆi + (a ⋅ ∇ )V ˆj + (a ⋅ ∇ )V ˆk 1 2 3

and

( ∂V1 ∂V1 ∂V1 ) ( ∂V2 ∂V2 ∂V2 ) ( ∂V3 ∂V3 ∂V3 ) (a ⋅ ∇ )V = a1 ----+ a2 ----+ a3---- ˆi+ a1 ----+ a2---- + a3---- ˆj+ a1 ----+ a2---- + a3---- ˆk ∂x ∂y ∂z ∂x ∂y ∂z ∂x ∂y ∂z

more to come..

This is from the public domain text by Gibbs.

"vector functions" is owned by bloftin.
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