Parametric Surfaces

Calculus 3 · Axiom Academy

Surfaces traced by a vector function of two parameters — their tangent vectors, the normal, and the surface-area element. 1. From Two Parameters to a Surface Two parameters u and v range over a domain D in the uv -plane — usually a rectangle. For each pair, returns a point (x,y,z) . As (u,v) sweeps the domain, those points sweep out a surface. 2. A First Example: The Cylinder Take a cylinder of radius a . Wrapping an angle around and rising a height z up gives a clean parametrization. Fixing z and varying traces a horizontal circle; fixing and varying z traces a vertical line. Together the two parameters lay a grid on the surface. 3. Tangent Vectors and the Normal Hold one parameter fixed and differentiate in the other: the result is a tangent vector to the surface, pointing along a grid curve. Tangent along the u -direction (holding v fixed). Tangent along the v -direction (holding u fixed). A tiny rectangle in the parameter plane maps to a small, nearly-flat patch on the surface, spanned by and . Adding up every patch over the domain gives the total surface area: 5. Worked Example: The Cylinder's Area Put every piece together for the cylinder . Because is constant, the lateral surface unrolls without stretching into a flat rectangle. Integrating over gives its area: describe complicated surfaces exactly, compute tangent planes and normal vectors, and find surface areas and surface integrals.

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