Engineering Calculus Notes 311

Engineering Calculus Notes 311 - z = t. The rst two...

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3.5. SURFACES AND THEIR TANGENT PLANES 299 is, neither is a scalar multiple of the other. The image of a regular parametrization S := { −→ p ( s,t ) | ( s,t ) dom( −→ p ) } is a surface in R 3 , and we will refer to −→ p ( s,t ) as a regular parametrization of S . As an example, you should verify (Exercise 9a ) that the graph of a (continuously diFerentiable) function f ( x,y ) is a surface parametrized by −→ p ( s,t ) = ( s,t,f ( s,t )) . As another example, consider the function −→ p ( θ,t ) = (cos θ, sin θ,t ); this can also be written x = cos θ y = sin
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Unformatted text preview: z = t. The rst two equations give a parametrization of the circle of radius one about the origin in the xy-plane, while the third moves such a circle vertically by t units: we see that this parametrizes a cylinder with axis the z-axis, of radius 1 (igure 3.17 ). x y z t igure 3.17: Parametrized Cylinder The partials are p ( ,t ) = (sin ) + (cos ) p t ( ,t ) = k...
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