Elliptical cylindrical (EC) coordinates are defined by the variables u, v and z, with a constant a, in terms of Cartesian coordinates as follows: x = a cosh u cos v ; y = a sinh u sin v ; (75) z = z . a. Find the EC metric. b. Describe surfaces of constant u, v and z for various values of the constant a. c. Find the EC basis vectors for arbitrary a. d. Make vector field plots of the EC basis vectors in the x-y plane. e. Find the incremental area and volume expressions. f. Express the gradient, divergence, curl and Laplacian operators in EC coordinates.

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Proper Length

The theory of special relativity explains the connection between space and time to objects that move in a straight line at a constant speed. The objects move at the speed of light. When an object approaches light speed, its mass is endless and cannot move faster than light. The most crucial postulate of Einstein's special relativity is that the laws of nature do not change in each frame of reference.