(a) Write down the line element for a three-dimensional Euclidean space using Cartesian coordinates, a', where...

 

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(a) Write down the line element for a three-dimensional Euclidean space using Cartesian coordinates, a', where i = 1, 2, 3. (b) Write down the metric [gy] and dual metric [g"] for this space, as arrays. (c) What value do all the connection coefficients take in this space? (d) Show that the geodesic equations in this metric lead to straight lines of the form r AX+ B where X is the affine parameter and A and B are constants. (a) Write down the line element for a three-dimensional Euclidean space using Cartesian coordinates, a', where i = 1, 2, 3. (b) Write down the metric [gy] and dual metric [g"] for this space, as arrays. (c) What value do all the connection coefficients take in this space? (d) Show that the geodesic equations in this metric lead to straight lines of the form r AX+ B where X is the affine parameter and A and B are constants.