Engineering Calculus Notes 380

Engineering Calculus Notes 380 - We saw in 3.2 that a...

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368 CHAPTER 3. REAL-VALUED FUNCTIONS: DIFFERENTIATION (b) f ( x,y ) = 3 x 2 + 10 xy 8 y 2 + 2 (c) f ( x,y ) = x 2 xy + y 2 + 3 x 2 y + 1 (d) f ( x,y ) = x 2 + 3 xy + y 2 + x y + 5 (e) f ( x,y ) = 5 x 2 2 xy + y 2 2 x 2 y + 25 (f) f ( x,y ) = 5 y 2 + 2 xy 2 x 4 y + 1 (g) f ( x,y ) = ( x 3 3 x )( y 2 1) (h) f ( x,y ) = x + y sin x Theory problems: 3. Show that the unit sphere S is a closed and bounded set. 4. (a) Mimic the proof given in the positive deFnite case of Proposition 3.8.3 to prove the negative deFnite case. (b) Prove Lemma 3.8.4 . 3.9 The Principal Axis Theorem In this section, we extend the analysis of quadratic forms from two to three variables, which requires some new ideas. ±irst, we need to clarify the mysterious “matrix representative” that appeared, for a quadratic form in two variables, in § 3.8 . Matrix Representative of a Quadratic Form
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Unformatted text preview: We saw in 3.2 that a linear real-valued function ( x ) can be expressed as multiplication of the coordinate column [ x ] of the input vector by a row of coecients; for R 2 , this reads ( x ) = b a 1 a 2 B x 1 x 2 = a 1 x 1 + a 2 x 2 = a 1 x + a 2 y while for R 3 it reads ( x ) = b a 1 a 2 a 3 B x 1 x 2 x 3 = a 1 x 1 + a 2 x 2 + a 3 x 3 . = a 1 x + a 2 y + a 3 z. Analogously, we can express any quadratic form as a three-factor product, using the basic matrix arithmetic which is reviewed in Appendix E . or...
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This note was uploaded on 10/20/2011 for the course MAC 2311 taught by Professor All during the Fall '08 term at University of Florida.

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