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m23n1s03ups10

# m23n1s03ups10 - MATH 23 Lecture Notes April 2010 NAME B...

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MATH 23 Lecture Notes April, 2010 NAME: B. Dodson We have the definition of the surface area of a surface S when S is given as a graph z = g ( x, y ) for ( x, y ) in a region D in the xy -plane, using the element of surface area dS. To match the treatment given in Chapter 16, we take r ( x, y ) = < x, y, g ( x, y ) >, and recall that dS was given as the product of the length of the cross product r x × r y , with dxdy since these vectors (scaled by dx and dy , respectively) span a parallelogram in the tangent plane whose area approximates the area of the small piece of the surface over a rectangle dxdy . Then the area of the (parameterized) surface is ZZ D | r x × r y | dA. See 16.6 formulas 8 and 9, and example 1 below. We use dS to define surface integrals RR S f dS . Next, we specify a unit normal n on S , and a vector field F = F ( x, y, z ), then the integral of F across S is given by ZZ S F dS = ZZ S F · n dS, which is the integral that appears in the two main formulas, Stokes’ Theorem and the Divergence formula. Two examples illustrate the integrals.

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m23n1s03ups10 - MATH 23 Lecture Notes April 2010 NAME B...

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