hw6-solutions

On the other hand if is indenite then the level

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Unformatted text preview: y (t)2 ) = c 2 for some constant c. Therefore, each solution curve is contained in a level curve of ψ . We claim that each level curve of the function ψ is an ellipse. Indeed, ψ (x, y ) is a quadratic form in x and y : ψ (x, y ) = 1 xy 2 a21 a22 a22 −a12 x . y If the quadratic form ψ is positive definite or negative definite, then the level curves of ψ are ellipses. On the other hand, if ψ is indefinite, then the level curves of ψ will be hyperbolas. In the present setting, we have det a21 a22 = −a21 a12 − a2 = −a21 a12 + a11 a22 > 0. 22 a22 −a12 This shows that the quadratic form ψ (x, y ) is either positive definite or negative definite. (It cannot be indefinite since it has positive determinant.) Therefore, each level curve of ψ is an ellipse. Secti...
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This note was uploaded on 02/09/2014 for the course MATH 53 taught by Professor Staff during the Winter '08 term at Stanford.

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