Space-invariant means homogeneous

[fisico30]

The concept of invariance of an object, property, etc... should always expresses with respect to something else:

time-invariant means static.
space-invariant means homogeneous.
direction invariant means isotropic.

"Something" can have just one of those types of invariance, all three of them, or none.

When talking about a vectors, a functions, formulas and equations I hear that they need to be invariant in the sense that they are independent of the coordinate system used to express them. A coordinate system is just a different way to describe the same phenomena (usually a simpler, more convenient way).

A vector, upon change of coord system, changes its description but its magnitude and true direction is always the same.

But when I think about equations, say the Helmholtz equation, when changing from Cartesian to spherical coord., the equation changes its functional form completely...
So what is actually invariant, what stays the same as far as equations goes?


We could also have invariance of objects with respect to a frame of reference, if the different frames are all inertial (moving at constant speed).
For ex: F=ma equation has the same form in another frame F'=ma'...


thanks
fisico30

 

[atyy]

Under Lorentz transformations, 4 vectors are invariant, but 3 vectors are not. E and B 3-vectors are not invariant, but the 4-potential is. Maxwell's equations in their usual form are also not "manifestly covariant", but they can be written in a different way so that their form does not change under a Lorentz transformation.

Take a look at Woodhouse's Lecture 12:
http://people.maths.ox.ac.uk/~nwoodh/sr/index.html

 

[Phrak]

But when I think about equations, say the Helmholtz equation, when changing from Cartesian to spherical coord., the equation changes its functional form completely...
So what is actually invariant, what stays the same as far as equations goes?

Box A is always zero.