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Engineering Calculus Notes 380

# Engineering Calculus Notes 380 - We saw in ย 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 definite case of Proposition 3.8.3 to prove the negative definite 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. First, 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 We saw in § 3.2
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Unformatted text preview: We saw in ยง 3.2 that a linear real-valued function โ ( โโ x ) can be expressed as multiplication of the coordinate column [ โโ x ] of the input vector by a row of coeยฒcients; for R 2 , this reads โ ( โโ x ) = b a 1 a 2 B ยท ยฑ x 1 x 2 ยฒ = a 1 ยท x 1 + a 2 x 2 = a 1 x + a 2 y while for R 3 it reads โ ( โโ x ) = b a 1 a 2 a 3 B ยท x 1 x 2 x 3 = a 1 ยท 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 . ยฑor...
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