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10.1. In lecture I explained that the expansion of the determinant of a matrix A has a term for every transversal of A, which is either added or subtracted depending on the choice of transversal. I also said that if some of the entries of A vanish, then some of the transversals do too, and those can be skipped in the expansion of detA. Let a, b = 0, and let A be an n X n matrix with a on the diagonal, with b just above the diagonal, and with b in the lower left corner, like this: a b a b a b a b a b a b a b b a b 0 0 0 a Describe the non-zero transversals of A and find detA. 10.2. Spherical coordinates for R³, minus the Z axis, are defined by: = Find the Jacobian matrix Dg for this coordinate system. As in problem 10.1, list the terms in the transversal expansion of detDg, skipping the terms that are always zero. Also calcu- late absolute Jacobian determinant to obtain the volume element for spherical coordinates, simplified as much as possible. (Hint: The determinant simplifies with repeated use of + cos² = 1.) 10.3. Let X C R² be the union of these two parametric curves: (x,y) = g(t)=(t,sin(3/t)) = t>1 0 (x,y)=h(t)=(0,t) = = X looks like this: 1 Prove that each curve is a valid coordinate chart, but that together they are not a valid atlas of charts for a 1-manifold. Specifically, explain rigorously why the conditions for an atlas of charts are not satisfied. (Note: X is not a 1-manifold and does not have a valid atlas of charts. But I am not asking you to prove that, only that this atlas doesn't work.) 10.4. 0 are two constants, we can define the following parametric surface M² in R³ with coordinates a and ß: (x,y,z)=g(a,B) =((b(cosß)+a)(cosa),((b(cosß)+a)(sina),b(sinß)) = Technically speaking we cannot make one chart this way because a and ß are periodic with period 2.. However, it is valid if we interpret it as an self-overlapping chart, as I will discuss in lecture. (a) When a is much larger than b, the torus M² can be called a "bicycle tire"; while if a and b are nearly equal, it can be called a "bagel". Draw diagrams with a and b labelled to justify these terms. (Hint: M² is also a surface of revolution of a circle.) (b) Use the volume formula ATA from lecture and from Munkres to calculate the area of M². (Hint: You can simplify using cos² + sin² = 1.)

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