Surface Area

Differential Geometry · Axiom Academy

LESSON Surface Area in Differential Geometry Understanding how to measure area on curved surfaces using the first fundamental form 1. Infinitesimal Parallelograms When we parametrize a surface as X(u,v) , small changes du and dv create infinitesimal tangent vectors that form a tiny parallelogram on the surface. The tangent vectors are X u du and X v dv , and the area of this parallelogram gives us our area element. 2. Area Element via Cross Product The area of a parallelogram spanned by two vectors is given by the magnitude of their cross product. This gives us our fundamental area element formula. Taking the cross product of the tangent vectors and computing its magnitude, we obtain the area element in terms of the parametrization. 3. Connection to First Fundamental Form The first fundamental form encodes how the surface metric works, with coefficients E = X u · X u , F = X u · X v , and G = X v · X v . Using the identity |a × b|² = |a|²|b|² - (a · b)², we can express the area element purely in terms of E, F, and G. To find the total surface area, we integrate the area element over the parameter domain. This double integral sums up all the infinitesimal parallelograms. The limits of integration depend on the parameter domain D in the uv-plane that covers the surface. For a sphere of radius R using spherical coordinates: X(θ,φ) = (R sin θ cos φ, R sin θ sin φ, R cos θ) Computing the tangent vectors and cross product: E = R², F = 0, G = R² sin² θ, giving dA = R² sin θ dθ dφ.

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