Engineering Calculus Notes 318

Engineering Calculus Notes 318 - ∂β has a nonzero...

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306 CHAPTER 3. REAL-VALUED FUNCTIONS: DIFFERENTIATION we see that the position vector of P is −−→ O P = −−→ O Q + −−→ QP = [( a cos α ) −→ ı + ( a sin α ) −→ ] + [( b cos β ) −→ v α + ( b sin β ) −→ k ] = [( a cos α ) −→ ı + ( a sin α ) −→ ] + ( b cos β )[(cos α ) −→ ı + (sin α ) −→ ] + ( b sin β ) −→ k so the torus (sketched in Figure 3.19 ) is parametrized by the vector-valued function −→ p ( α,β ) = ( a + b cos β )[(cos α ) −→ ı + (sin α ) −→ ] + ( b sin β ) −→ k (3.26) Figure 3.19: Torus The partial derivatives of this function are −→ p ∂α = ( a + b cos β )[( sin α ) −→ ı + (cos α ) −→ ] −→ p ∂β = ( b sin β )[(cos α ) −→ ı + (sin α ) −→ ] + ( b cos β ) −→ k . To see that these are independent, we note ±rst that if cos β n = 0 this is obvious, since −→ p
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Unformatted text preview: ∂β has a nonzero vertical component while ∂ −→ p ∂α does not. If cos β = 0, we simply note that the two partial derivative vectors are perpendicular to each other (in fact, in retrospect, this is true whatever value β has). Thus, every point is a regular point. Of course, increasing either α or β by 2 π will put us at the same position, so to get a coordinate patch we need to restrict each of our parameters to intervals of length < 2 π . To de±ne the tangent plane to a regularly parametrized surface, we can think, as we did for the graph of a function, in terms of slicing the surface...
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