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

[Rosicky08]

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

 

[coalquay404]

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

 

[christianjb]

How long is a piece of string?

 

[pmb_phy]

There is

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

Pete

 

[masudr]

How long is a piece of string?

Forty two.

 

[christianjb]

Forty two.

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

 

[HallsofIvy]

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 anyone 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.

 

[pmb_phy]

Forty two.

Well said.