Engineering Calculus Notes 386

# Engineering Calculus Notes 386 - 374 CHAPTER 3 REAL-VALUED...

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374 CHAPTER 3. REAL-VALUED FUNCTIONS: DIFFERENTIATION and clearly −→ g 2 ( −→ u ) = −→ u 1 , so the frst equation reads 2 A −→ u = 2 λ −→ u + μ −→ u 1 . IF we take the dot product oF both sides oF this with −→ u 1 , using the Fact that −→ u · −→ u 1 = 0 and −→ u 1 · −→ u 1 = 1, we obtain 0 = μ so For −→ u = −→ u i , i = 2 , 3, the frst equation, as beFore, is the eigenvector condition A −→ u i = λ i −→ u i . We already know that −→ u 1 · −→ u 2 = −→ u 1 · −→ u 3 = 0 and −→ u 2 · −→ u 2 = −→ u 3 · −→ u 3 = 1, but what about −→ u 2 · −→ u 3 ? IF Q is constant on P ∩ S 2 , then every vector in P ∩ S 2 qualifes as −→ u 2 and/or −→ u 3 , so we simply pick these to be mutually perpendicular. IF not, then λ 2 = −→ u 2 · ( λ 2 −→ u 2 ) = Q ( −→ u 2 ) = min −→ u ∈P∩S 2 Q ( −→ u ) < max −→ u ∈P∩S
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