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Engineering Calculus Notes 310

Engineering Calculus Notes 310 - immediately with...

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298 CHAPTER 3. REAL-VALUED FUNCTIONS: DIFFERENTIATION giving as the equation of the tangent plane 3( x + 2) + 58( y 3) + 18( z 1) = 0 or 3 x + 58 y + 18 z = 186 . You should check that this is equivalent to any one of the forms of the equation given in Proposition 3.5.5 . Parametrized Surfaces In § 2.2 we saw how to go beyond graphs of real-valued functions of a real variable to express more general curves as images of vector-valued functions of a real variable. In this subsection, we will explore the analogous representation of a surface in space as the image of a vector-valued function of two variables. Of course, we have already seen such a representation for planes. Just as continuity and limits for functions of several variables present new subtleties compared to their single-variable cousins, an attempt to formulate the idea of a “surface” in R 3 using only continuity notions will encounter a number of difficulties. We shall avoid these by starting out
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Unformatted text preview: immediately with di±erentiable parametrizations. Defnition 3.5.6. A vector-valued function −→ p ( s,t ) = ( x 1 ( s,t ) ,x 2 ( s,t ) ,x 3 ( s,t )) of two real variables is diferentiable (resp. continuously diferentiable , or C 1 ) if each of the coordinate functions x j : R 2 → R is diFerentiable (resp. continuously diFerentiable). We know from Theorem 3.3.4 that a C 1 function is automatically diFerentiable. We de±ne the partial derivatives of a diFerentiable function −→ p ( s,t ) to be the vectors ∂ −→ p ∂s = p ∂x 1 ∂s , ∂x 2 ∂s , ∂x 3 ∂s P ∂ −→ p ∂t = p ∂x 1 ∂t , ∂x 2 ∂t , ∂x 3 ∂t P . We will call −→ p ( s,t ) regular if it is C 1 and at every pair of parameter values ( s,t ) in the domain of −→ p the partials are linearly independent—that...
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