Surface Integral of Scalar Function

Calculus 3 · Axiom Academy

LESSON Surface Integrals of Scalar Functions Integrating a scalar field over a curved surface — how the area element dS turns a flat patch into real area, mass, and more. 1. Summing a Field Over a Surface Take a surface S and a scalar field f living on it — say a density (mass per unit area). Chop S into tiny patches, multiply each patch's area dS by the field value there, and add. The limit of that sum is the scalar surface integral . With (density) this is the total mass of the shell With f=1 every patch just contributes its area — the total is the surface area To compute anything we need coordinates on the surface. A parametrization maps a flat region D in the uv -plane onto the curved surface S — carrying a grid of squares to a warped grid of curved cells. as (u,v) ranges over the domain D . For a hemisphere of radius R we can use spherical angles: Near a point, the tangent vectors and span the surface. A domain rectangle of sides du,dv maps to the parallelogram they span, whose area is the length of their cross product. That length is the stretch factor . For a graph z=g(x,y) this becomes a clean square root: 4. Independent of Parametrization and Orientation The value of does not depend on which parametrization you pick. Reparametrize with and the Jacobian of the change of variables exactly cancels the change in : 5. Worked Example: Mass of a Hemisphere

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