Engineering Calculus Notes 325

Engineering Calculus Notes 325 - 313 3.5 SURFACES AND THEIR...

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Unformatted text preview: 313 3.5. SURFACES AND THEIR TANGENT PLANES The partials are → ∂− p π− π → + cos π − → − sin ı = 2 + cos ∂α 3 6 6 √ 3− 1 1→ → = 2+ − −+ ı 2 2 2 √ 5− → →+ 5 3 − =− ı 4 4 → → π− π ∂− p → + sin π − + cos π − → cos ı k = 2 − sin ∂β 3 6 6 3 √ √ → 3 3− → + 1− + 1− → = 2− k ı 2 2 2 2 √ √ → 3− → → + 1 − 3 − + 1− 3− = k ı 4 4 2 so a parametrization of the tangent plane is √ √ 3 5 53 3− − △α + △β x= 4 4 4 √ √ 53 3 5 △α + 1 − △β y= + 4 4 4 √ 1 3 z= △α + △β. 2 2 The normal to the tangent plane is → − N= = − 5 4 √ 53 8 → −+ ı → −+ ı √ 53 4 5 8 → − → −+ so an equation for the plane is √ √ 5 53 53 x− + 8 4 8 Exercises for § 3.5 × √ 3 3− 4 → −+ ı 1− √ 3 4 → → − + 1− k 2 √ → − 65 3 −5 k 16 5 y− 4 + √ 65 3 −5 16 z− √ 3 2 = 0. ...
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