WHY there is no scalar equation for a line in three dimensions ?

1. Apr 13, 2007

Rosicky08

WHY there is no scalar equation for a line in three dimensions ???

2. Apr 13, 2007

coalquay404

Perhaps you might like to try rephrasing that in a manner which is understandable.

3. Apr 13, 2007

christianjb

How long is a piece of string?

4. Apr 13, 2007

pmb_phy

There is

ds2 = $\eta_{\alpha\beta}$ dx$\alpha$ dx$\beta$

Pete

Last edited: Apr 13, 2007
5. Apr 18, 2007

masudr

Forty two.

6. Apr 18, 2007

christianjb

All pieces of string are of length 42, just in different units.

7. Apr 18, 2007

HallsofIvy

Staff Emeritus
Rosicky08, R3 has three dimensions. A curve has one dimension so you have to "lose" 2 dimensions. If you had a single, scalar equation, in three variables, you could solve for any one in terms of the other TWO. That gives you "two degrees of freedom": pick any two coordinates and solve for the third: that one equation allows you to "lose" on dimension. Any single scalar equation describes a two dimensional surface. In order to get one dimension you must have either 2 equations in the three space coordinates so that you could solve for two variables in terms of 1 or 3 equations in the 3 space coordinates and one parameter: 3 equations in 4 variables gives you 4-3= 1 degree of freedom or one dimension.

8. Apr 18, 2007

pmb_phy

Well said. :rofl: