11.23 - Section 13.5 wrap-up[Show examples on pages 961 and...

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Section 13.5 wrap-up: [Show examples on pages 961 and 962.] Section 13.6: Parametric surfaces and their areas Key ideas? ..?.. Parametric surfaces and grid-lines How the form and/or symmetry of a surface helps one in choosing a parametrization Differentiability and tangent planes to parametric surfaces Surface area: If S = { r ( u , v ): ( u , v ) in D }, then A ( S ) = ∫∫ D | r u × r v | dA Why parametrize a surface? ..?.. It lets us plot surfaces more easily and compute associated quantities such as surface area. Individual drill: Parametrize a cylinder (i.e. cylindrical surface) of radius 2 with axis the z -axis. ..?..
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One possibility: [2 cos θ , 2 sin θ , z ]. Although one can use any variables to parametrize a surface, we’ll frequently use u and v . Sometimes we parametrize a surface by parametrizing separate parts of it and pasting them together. Example: Let’s parametrize the y 0 half of the ellipsoid x 2 /4 + y 2 + z 2 /4 = 1. x 2 + z 2 = 4(1– y 2 ), so if we let u = 2 sqrt(1– y 2 ), then x 2 + z 2 = u 2 .
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