# Stanford University Derivatives Problems Analysis Real Analysis proof, Need the strict proof of these real analysis question. Thank you a 10.1. In lecture

Stanford University Derivatives Problems Analysis Real Analysis proof, Need the strict proof of these real analysis question. Thank you a
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 0 0]
[a b 0 0 0]
0 а ь о 0
0 a b 0
| oo a bo
oo a b
000 a b
b 0 0 a
b 0 0 0
Describe the non-zero transversals of A and find detA.
10.2. Spherical coordinates for R}, minus the z axis, are defined by:
(x,y,z) = (r, 0,0) = (r(sin )(coso), r(sin ) (sin o), r(cos O)).
Find the Jacobian matrix Dg for this coordinate system. As in problem 10.1, list the terms
in the transversal expansion of det Dg, 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
sin? +cos= 1.)
10.3. Let X CR2 be the union of these two parametric curves:
(x,y) = g(t) =(t, sin(3/t))
t>O
(x,y)=h(t) = (0,1)
t