Word Count : 189

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To solve this problem, we need to determine the metric coefficients for the coordinate system \((u, v, z)\) whose coordinates are related to the Cartesian coordinates \((x, y, z)\) by the following equations: \[ x = \frac{1}{2}(u^2 _____ _________ ____ _______ ______ _______ __________ _____ _________ _______ ______ _________.
________ ________ __________ ___ __________.
_______ ______ __________ _________ ________ _____.
___ ________ ____ _____ ______ _________ _______ ________ ___ _______ _________.
__________ ___ ____ _________ ___ ____ _______.
________ ________ _____ ______ __________ _________ ________ ___ ______.
_________ _________ ________ ________ ________ ________ ____ ___.
_______ __________ ______ ______ ___.
____ ___ _______ __________ _______ _________ __________ ___.
_______ _______ ___ ________ ___ __________ _____ __________.
_________ _________ ____ ________ ________ _______ _______ __________ ________.
___ ______ ______ ___ ___ _____ ______.
______ _________ __________ ___ _____ _____ ____ ___ _______ _____.
___ _______ ________ __________ _______ ___ ________.
________ _________ _______ ______ _____ _________ ________.
_____ ______ _________ ___ _____ _____ _____ _________ ___ ______ _____.
__________ _______ ____ ______ ________ _____ ________ ___.
______ ___ ______ _______ ______ _________ _______ ___ ____ ________ _________ ___.
__________ __________.
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