Questions: The First Fundamental Form of a Surface

Question 1

For the sphere x(φ, θ) = (R sin φ cos θ, R sin φ sin θ, R cos φ), what are E, F and G?

• A. E = R², F = 0, G = R² sin²φ
• B. E = R² sin²φ, F = 0, G = R²
• C. E = R, F = R, G = R sin φ
• D. E = 1, F = 0, G = 1

Answer: (A)

x_φ = (R cos φ cos θ, R cos φ sin θ, -R sin φ) has squared length R², x_θ = (-R sin φ sin θ, R sin φ cos θ, 0) has squared length R² sin²φ, and their dot product is 0. So the meridians and parallels are perpendicular and E = R², F = 0, G = R² sin²φ.

Question 2

Using the sphere's coefficients E = R², F = 0, G = R² sin²φ, compute the total surface area of the sphere.

Model answer: 4πR²

The area element is √(EG - F²) dφ dθ = R² sin φ dφ dθ. Integrating over 0 ≤ φ ≤ π and 0 ≤ θ ≤ 2π gives R² · 2 · 2π = 4πR².

Question 3

The cylinder x(θ, z) = (cos θ, sin θ, z) and the plane x(u, v) = (u, v, 0) have the same first fundamental form in these coordinates.

• T. True
• F. False

Answer: True

For the cylinder, x_θ = (-sin θ, cos θ, 0) and x_z = (0, 0, 1), so E = 1, F = 0, G = 1, exactly as for the plane. This is why a flat sheet can be rolled into a cylinder without stretching: small regions of the two surfaces are isometric.

Question 4

Write the formula for the length of a curve t ↦ x(u(t), v(t)) on a surface, for a ≤ t ≤ b, in terms of E, F and G.

Model answer: Length = ∫ from a to b of √(E u'(t)² + 2F u'(t) v'(t) + G v'(t)²) dt

The velocity is x_u u' + x_v v' by the chain rule. Its squared length is (x_u · x_u)u'² + 2(x_u · x_v)u'v' + (x_v · x_v)v'², which is the first fundamental form evaluated on (u', v').