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

Click For Summary
A scalar equation for a line in three dimensions is not possible because it would describe a two-dimensional surface instead. In three-dimensional space, a line is one-dimensional, requiring the loss of two dimensions to represent it. To achieve this, either two equations must be used to solve for two variables in terms of one, or three equations must be established in four variables to yield one degree of freedom. The discussion emphasizes that a single scalar equation cannot capture the necessary dimensionality of a line. Thus, multiple equations are essential for accurately defining a line in three-dimensional space.
Rosicky08
Messages
1
Reaction score
0
WHY there is no scalar equation for a line in three dimensions ?

:confused:
 
Physics news on Phys.org
Perhaps you might like to try rephrasing that in a manner which is understandable.
 
How long is a piece of string?
 
Rosicky08 said:
:confused:
There is

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

Pete
 
Last edited:
christianjb said:
How long is a piece of string?

Forty two.
 
masudr said:
Forty two.

All pieces of string are of length 42, just in different units.
 
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.
 
masudr said:
Forty two.
Well said. :smile:
 

Similar threads

Undergrad Color charge is not scalar -- do their components have dimensions?
  • · Replies 12 ·
Replies
12
Views
1K
Undergrad Differential structure on topological manifolds of dimension <= 3
  • · Replies 11 ·
Replies
11
Views
3K
High School What is the definition of a straight line in mathematics?
  • · Replies 7 ·
Replies
7
Views
2K
High School Can Curvature and Geodesics be Generalized in Higher Dimensions?
  • · Replies 10 ·
Replies
10
Views
3K
Undergrad Meaning of cross terms in line element
  • · Replies 1 ·
Replies
1
Views
3K
Undergrad Inner product of two tensors
  • · Replies 6 ·
Replies
6
Views
2K
Graduate Why Does the Standard Model Use Six Dimensions for Probability Calculations?
  • · Replies 20 ·
Replies
20
Views
2K
Graduate Renormalizability conditions for a real scalar field in d dimensions
  • · Replies 2 ·
Replies
2
Views
2K
Maxwell's Equations in N Dimensions
  • · Replies 10 ·
Replies
10
Views
3K
High School Is velocity ever a scalar quantity?
  • · Replies 48 ·
2
Replies
48
Views
6K