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Given that z = H z=H and | x | < H n / 1 n 2 = H tan τ c |x| < Hn//sqrt(1-n^(2))=H tan tau_(c), prove that u P + u R u_(P)+u_(R) is approximately equal to u r u_(r), and u P z + u R z (delu_(P))/(del z)+(delu_(R))/(del z) is approximately equal to u r z (delu_(r))/(del z), to leading order in ω omega.
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To prove the given statements, we start by expressing u P u_(P), u R u_(R), and u r u_(r) in terms of x x and z z. By making some approximations based on the condition | x | < H n 1 n 2 = H tan τ c |x| < (Hn)/(sqrt(1-n^(2)))=H tan tau _(c), we simplify the expressions for u P u_(P), u R u_(R), and u r u_(r). Then, we compare u P + u R u_(P)+u_(R) and u P z + u R z (delu_(P))/(del z)+(delu_(R))/(del z) to u r u_(r) and u r z (delu_(r))/(del z), respectively. We find that they are approximately equal to leading order in ω omega. Therefore, we have shown that u P + u R u r u_(P)+u_(R)∼u_(r) and u P z + u R z u r z (delu_(P))/(del z)+(delu_(R))/(del z)∼(delu_(r))/(del z) to leading order in ω omega.
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