Surfaces Summary

Differential Geometry · Axiom Academy

Unit 2: Key concepts in differential geometry of surfaces Definition: A surface is regular when its parametrization X(u,v) is smooth and has linearly independent partial derivatives at every point Regularity Condition: The vectors X u and X v must be non-zero and non-parallel Why It Matters: Regular surfaces have well-defined tangent planes and normal vectors at every point, making calculus possible Test: Check that the cross product X u × X v ≠ 0 everywhere Tangent Plane: At each point, the plane spanned by the partial derivative vectors X u and X v Normal Vector: The vector perpendicular to the tangent plane, found via N = X u × X v Unit Normal: Normalized version n = N / ||N|| has length 1 Geometric Meaning: The normal vector points "straight out" from the surface and is essential for measuring curvature First Fundamental Form Coefficients E coefficient: Measures stretching in the u -direction: F coefficient: Measures interaction between u and v directions: G coefficient: Measures stretching in the v -direction: Meaning: These encode all intrinsic distance measurements on the surface Area Formula: Integrate over the parameter domain Why the square root?: The expression EG - F² equals ||X u × X v ||² , the squared area of the parallelogram Practical Use: Convert surface integrals to double integrals in parameter space Connection: Generalizes the 2D area element dx dy to curved surfaces Example: How Curves Live on Surfaces

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