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Engineering Calculus Notes 382

Engineering Calculus Notes 382 - 370 CHAPTER 3 REAL-VALUED...

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370 CHAPTER 3. REAL-VALUED FUNCTIONS: DIFFERENTIATION are the coefficients a ii of the “square” terms x 2 i , and each off-diagonal entry is half of the coefficient b ij of a “mixed product” term x i x j : if Q ( x 1 ,x 2 ,x 3 ) = a 11 x 2 1 + b 12 x 1 x 2 + b 13 x 1 x 3 + a 22 x 2 2 + b 23 x 2 x 3 + a 33 x 2 3 then we rewrite it in “balanced” form Q ( x 1 ,x 2 ,x 3 ) = a 11 x 2 1 + a 12 x 1 x 2 + a 13 x 1 x 3 + a 21 x 2 x 1 + a 22 x 2 2 + a 23 x 2 x 3 + a 31 x 3 x 1 + a 32 x 3 x 2 + a 33 x 2 3 where a 12 = a 21 = 1 2 b 12 a 13 = a 31 = 1 2 b 13 a 23 = a 32 = 1 2 b 23 and its matrix representative is [ Q ] = a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 = a 11 1 2 b 12 1 2 b 13 1 2 b 12 a 22 1 2 b 23 1 2 b 13 1 2 b 23 a 33 . The Principal Axis Theorem Using the language of matrices, Proposition 3.8.5 can be rephrased as:
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