265L21 - LESSON 21 Parametric Surfaces and Surface...

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LESSON 21 Parametric Surfaces and Surface Integrals READ: Section 17.4 NOTES: There is a two dimensional analog of line integrals called surface integrals where the region of integration is a surface rather than a curve. Just as parametric representations of curves are used to compute line integrals, surfaces will often be represented parametrically to do surface integrals. One way to describe a surface is with an explicit function z = f ( x,y ). For example, z = x 2 + 2 y 2 is an explicit representation of an elliptic paraboloid. Not every surface can be described with an explicit function. For example, the surface of the sphere of radius 1 centered at the origin is described by the equation x 2 + y 2 + z 2 = 1, but of course that cannot be solved for z to give an explicit representation since z is not a function of x and y in this case. To get explicit representations of the surface of the sphere, it would be necessary to look at two separate equations: z = p 1 - x 2 - y 2 gives the top half of the sphere and z = - p 1 - x 2 - y 2 gives the bottom half of the sphere. There is a second way to describe a surface that can be more convenient than an explicit representation. In a parametric representation of a surface, there will be three equations which describe x , y , and z in terms of two parameters u and v : x = x ( u,v ) y = y ( u,v ) z = z ( u,v ) . The variables u and v take on values in some region D of the u , v -plane, and the corresponding points ( x,y,z ) trace out a surface in 3-space. This parametric representation can also be expressed in vector form as -→ r ( u,v ) = x ( u,v ) -→ ı + y ( u,v ) -→ + z ( u,v ) -→ k , ( u,v ) D. As for the parameterization of a curve, a surface can be parameterized in many different ways. Most of the formulas and computations with surface integrals will be expressed in terms of a vector equation for the surface. If a surface is given by an explicit function, it is always possible to produce a paramet- ric equation for the surface in a simple fashion. Suppose the surface has equation z = f ( x,y ). For the parameters, we will use x and y themselves. Then the surface has vector equation -→ r ( x,y ) = x -→ ı + y -→ + f ( x,y ) -→ k . For surfaces not given by an explicit function, it will be necessary to select two parameters that can be used
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265L21 - LESSON 21 Parametric Surfaces and Surface...

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