**Parameterizing an ellipsoid** I'm working on the parametric representation for the lower half of the ellipsoid defined by 2x² + 7y² + z² = 1, which means z ≤ 0. I think I should use a common parameterization similar to spherical coordinates. So, I set x, y, and z based on scaling factors from the standard form. After re-examining the equation, I find that the parameters a, b, and c correspond to the correct values. Now, I just need to confirm the ranges for φ and θ.**Defining parameters for the ellipsoid** Since we're focusing on the lower half of the ellipsoid where z ≤ 0, that means cos φ is non-positive, so I set φ within [π/2, π]. For parameters u and v, I could choose u in [0, 2π) and v in [0, 1] or maybe use v in [π/2, π]. I might also explore elliptical coordinates where x and y are defined inside the ellipse. If I set r equal to sin v, I can express z as -cos v, so that could be helpful too!**Verifying the parameterization** I'm realizing that for the parameterization of the ellipsoid, x and y need different scalings. I set x = a sin v cos u and y = b sin v sin u, using a = 1/√2 and b = 1/√7. When I check the calculations, everything adds up perfectly. The sum gives 1, which works! Since we're exploring the lower half, I'll go with z = -cos v and restrict v to the range of [0, π/2]. That seems to be the more standard approach!**Choosing the right parameterization** I see that the problem is asking for a parametric representation of the lower half of the ellipsoid defined by 2x² + 7y² + z² = 1. I'm realizing that while u is typically in [0, 2π) and v in [0, π], I'll need to restrict v to [π/2, π] for this case. A solid parameterization could be r(u,v) = ((1/√2) sin v cos u, (1/√7) sin v sin u, -cos v) with u in that standard range. I think this spherical-like approach is safe and effective!**Clarifying parameterization details** I need to ensure that I clearly communicate the domains for u and v in a straightforward way. I’ll explain the general formula: x = a sin v cos u, y = b sin v sin u, and z = -cos v, with a = 1/√2 and b = 1/√7. Additionally, I can suggest an alternative parameterization using u in [0, 2π) and v in [0, 1]. Since the prompt requests equations for x, y, and z based on u and/or v, I'll present them clearly while keeping in mind that we're focusing on the lower half where z ≤ 0. Let’s summarize the final results succinctly!