CALCULUS FINAL EXAM - NTU 2011 CHIN-LUNG WANG
JUNE 16, PM 12:30 - 3:15
1.Consider the function U=F(X) = (x2−y2, xy).
(a) Obtain an iterative approximation G(X), which depends on given U, for the inverse transformation F−1(U)near X0 = (1, 1)or U0 = (0, 1). Verify that the fixed point Xfixedof G satisfies U=F(Xfixed).
(b) Show that there exists a δ>0 s.t. for any U∈Bδ(U0)the iteration Xn+1= G(Xn)with initial value X0converges to a limit, denoted by X(U). 2.Evaluate the integrals:
(a) Z 1
0 Z 1
y ex2dx dy, (b) Z 1
0 Z
√ 1−z2 0
Z
√1−y2−z2
0 (x2+y2+z2)xyz dx dy dz.
3.Evaluate the integral Z
{x2+y2+z2≤1}ex+y+zdx dy dz.
4.Evaluate the improper integral Z ∞
0
e−bx−e−ax
x cos x dx.
(State explicitly the theorem you use and check all the required conditions.)
5.Calculate Z
Sz dx∧dy−x dy∧dz,
where S is the spherical cap x2+y2+z2 = 1, x > 1/2, oriented positively with respect to the normal pointing to infinity.
6. Prove Green’s theorem onR2. Use it to derive one of the following: (1) The change of variable formula of double integrals with a C2transformation. (2) Stoke’s theorem for oriented surface S with ∂S=C inR3.
7. Derive the formula for Laplace operator onR3in spherical coordinates. Use it to show that the only radial vector field F (i.e. F(r) = a(r)rfor some function a in r= |r|) with curl F=0 and div F=0, except possibility at r=0, is given by
F= cr r3.
8. It is known experimentally that a charge conducting spherical lamina exerts zero force on a point charge inside the sphere. Assume that point charges repel or attract each other with a force dependent only only the distance between them, prove that this experiment implies Columb’s law.
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