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Uploaded on May 29, 2009
We look at a quantitative description of curves in space in terms of time dependent position vectors. We find a formula for the arc length of a curve or trajectory. We use the invariance of the scalar product to write the formula in terms of the metric tensor g(i,j). This leads to the famous differential form of the arc length: ds^2 = g(i,j) dx(i) dx(j). These formulas are general and work in higher dimensions.