plugin-Math255-Pset8

# plugin-Math255-Pset8 - 1 Winter 2010 Math 255 and bounded...

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Winter 2010 Math 255 Problem Set 8 Due Tuesday 9 March Section 15.6: 6, 10, 30, 60, 61 Section 15.7: 2, 18 (Graph the function), 36, 52, 53 Section 15.8: 2, 18, 22, 38, 43 (Give a clear argument) Challenge problem : a) Let u be a twice continuously diﬀerentiable func- tion on D which is “strictly subharmonic”: that is, the following inequality holds: 2 u = 2 u ∂x 2 + 2 u ∂y 2 > 0. Show that u cannot have a maximum point in D/∂D (the set of points in D but not on the boundary of D ). bi) Consider the function f ( x 1 ,x 2 ,x 3 ) = 1 2 3 X i =1 3 X j =1 a i,j x i x j with a ij = a ji 6 = 0. What is f ? Remark For those with some linear algebra background, f ( x ) = 1 2 ( Ax ) · x where A is a nonzero symmetric 3 × 3 matrix with entries a i,j and x = [ x 1 ,x 2 ,x 3 ]. bii) (extra credit) Consider the restriction of f to the unit sphere S = { ( x 1 ,x 2 ,x 3 ) | x 2 1 + x 2 2 + x 2 3 = 1 } in R 3 . The function f is continuous and is deﬁned on a closed

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Unformatted text preview: 1 Winter 2010 Math 255 and bounded domain, S , and so has a maximum and minimum on the domain S , the unit sphere. Show that there must be x 1 , x 2 and x 3 , with x 2 1 + x 2 2 + x 2 3 = 1 and a scalar λ 6 = 0 such that for i = 1, 2 and 3, 3 X j =1 a i,j x j = λx i . Remark For those with some linear algebra background, the vector x is called an eigenvector, while the scalar λ is called an eigenvalue. biii) What are the maxima and minima for f on B = { ( x 1 ,x 2 ,x 3 ) | x 2 1 + x 2 2 + x 2 3 ≤ 1 } ? Review problem (extra credit) : Give a rigorous proof of the chain rule for a scalar function of one variable. Please give all details and as-sumptions. 2...
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plugin-Math255-Pset8 - 1 Winter 2010 Math 255 and bounded...

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