x(u, v) = a cos(u) sin(v) y(u, v) = b sin(u) sin(v) z(u, v) = c cos(v)

x^2/a^2 + y^2/b^2 + z^2/c^2 = 1

2

g = 0

S = 2 π (c^2 + b sqrt(a^2 - c^2) E(am(sn^(-1)(sqrt(a^2 - c^2)/a|(a^2 (b^2 - c^2))/(b^2 (a^2 - c^2))), (a^2 (b^2 - c^2))/(b^2 (a^2 - c^2)))|(a^2 (b^2 - c^2))/(b^2 (a^2 - c^2))) + (b c^2 sn^(-1)(sqrt(a^2 - c^2)/a|(a^2 (b^2 - c^2))/(b^2 (a^2 - c^2))))/sqrt(a^2 - c^2))

ds^2 = sin^2(v) (a^2 sin^2(u) + b^2 cos^2(u)) du^2 + 2 (b^2 - a^2) sin(u) cos(u) sin(v) cos(v) du dv + cos^2(v) (a^2 cos^2(u) + b^2 sin^2(u)) + c^2 sin^2(v) dv^2

dA = sin(v) sqrt(c^2 sin^2(v) (a^2 sin^2(u) + b^2 cos^2(u)) + a^2 b^2 cos^2(v)) du dv

x^_ = (0, 0, 0)

V = 4/3 π a b c

I = (1/5 (b^2 + c^2) | 0 | 0 0 | 1/5 (a^2 + c^2) | 0 0 | 0 | 1/5 (a^2 + b^2))

K(u, v) = (a^2 b^2 c^2)/(c^2 sin^2(v) (a^2 sin^2(u) + b^2 cos^2(u)) + a^2 b^2 cos^2(v))^2

(for an ellipsoid with center at the origin and semi-axes a, b, and c lying along the Cartesian axes)

g_(uu) = sin^2(v) (a^2 sin^2(u) + b^2 cos^2(u)) g_(uv) = (b^2 - a^2) sin(u) cos(u) sin(v) cos(v) g_(vu) = (b^2 - a^2) sin(u) cos(u) sin(v) cos(v) g_(vv) = cos^2(v) (a^2 cos^2(u) + b^2 sin^2(u)) + c^2 sin^2(v)

Γ | u | | | uu = ((a^2 - b^2) sin(u) cos(u))/(a^2 sin^2(u) + b^2 cos^2(u)) Γ | u | | | uv = cot(v) Γ | u | | | vu = cot(v) Γ | u | | | vv = ((a^2 - b^2) sin(u) cos(u))/(a^2 sin^2(u) + b^2 cos^2(u)) Γ | v | | | uu = -(sin(v) cos(v) (a^2 cos^2(u) + b^2 sin^2(u)))/(a^2 cos^2(u) cos^2(v) + b^2 sin^2(u) cos^2(v) + c^2 sin^2(v)) Γ | v | | | uv = ((b^2 - a^2) sin(u) cos(u) cos^2(v))/(a^2 cos^2(u) cos^2(v) + b^2 sin^2(u) cos^2(v) + c^2 sin^2(v)) Γ | v | | | vu = ((b^2 - a^2) sin(u) cos(u) cos^2(v))/(a^2 cos^2(u) cos^2(v) + b^2 sin^2(u) cos^2(v) + c^2 sin^2(v)) Γ | v | | | vv = (sin(v) cos(v) (a^2 (-cos^2(u)) - b^2 sin^2(u) + c^2))/(a^2 cos^2(u) cos^2(v) + b^2 sin^2(u) cos^2(v) + c^2 sin^2(v))

E(u, v) = sin^2(v) (a^2 sin^2(u) + b^2 cos^2(u)) F(u, v) = (b^2 - a^2) sin(u) cos(u) sin(v) cos(v) G(u, v) = cos^2(v) (a^2 cos^2(u) + b^2 sin^2(u)) + c^2 sin^2(v)

e(u, v) = (a b c sin^2(v))/sqrt(a^2 b^2 cos^2(v) + a^2 c^2 sin^2(u) sin^2(v) + b^2 c^2 cos^2(u) sin^2(v)) f(u, v) = 0 g(u, v) = (a b c)/sqrt(a^2 b^2 cos^2(v) + a^2 c^2 sin^2(u) sin^2(v) + b^2 c^2 cos^2(u) sin^2(v))

left double bracketing bar x(u, v) right double bracketing bar = sqrt(sin^2(v) (a^2 cos^2(u) + b^2 sin^2(u)) + c^2 cos^2(v))

N^^(u, v) = ((cos(u) sin(v) b c)/sqrt(cos^2(v) a^2 b^2 + sin^2(v) (sin^2(u) a^2 + cos^2(u) b^2) c^2), (sin(u) sin(v) a c)/sqrt(cos^2(v) a^2 b^2 + sin^2(v) (sin^2(u) a^2 + cos^2(u) b^2) c^2), (cos(v) a b)/sqrt(cos^2(v) a^2 b^2 + sin^2(v) (sin^2(u) a^2 + cos^2(u) b^2) c^2))

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