FINAL EXAM OF CALCULUS
Date: 2000, June 19, 13:10–14:55
An answer without reasoning will not be accepted.
1. [8%] Find curl F and div F if F(x, y, z) = x2zi + 2x sin yj + 2z cos yk.
2. [8%] Show that there is no vector field G such that curl G = 2xi + 3yzj − xz2k.
3. [10%] Compute the outward flux of F(x, y, z) = xi+yj+zk through the ellipsoid 4x2+9y2+6z2= 36.
4. [10%] Compute the surface integral Z Z
S
(x2+ y2) dσ, where S is the hemisphere z =p
1 − (x2+ y2).
5. [14%] Let Ω be a Jordan region (on the xy-plane) with a piece-wise smooth boundary C, and let f and g be continuously differentiable functions on an open set containing Ω.
(i) Use the vector form of the Green’s theorem to prove Green first identity:
Z Z
Ω
f ∇2g dσ = I
C
f (∇g) · n ds − Z Z
Ω
∇f · ∇g dσ where n is the outer unit normal vector.
(ii) Use above Green’s first identity to prove Green’s second identity:
Z Z
Ω
(f ∇2g − g∇2f ) dσ = I
C
(f ∇g − g∇f ) · n ds.
6. [12%] Show that the vector field v(x, y, z) = (2xz +sin y)i+x cos yj+x2k is a gradient. Then evaluate the line integral of v over the curve r(u) = cos ui + sin uj + uk for u ∈ [0, π].
7. [10%] Find the volume of the largest rectangular box with edges parallel to the axes that can be inscribed in the ellipsoid x2+ 4y2+ z2= 36.
8. [6%] Compute the sum of the series
1 − ln π +(ln π)2
2! −(ln π)3 3! + · · ·
9. [8%] Compute the interval of convergence of the Taylor series in x of ln(1 − x).
10. [8%] Find the tangential component of the acceleration vector of r(t) = cos ti + sin tj + tk.
11. [8%] A function f (x, y) is called homogeneous of degree n if f has continuous second-order partial derivatives and f (tx, ty) = tnf (x, y) for all t. Use the chain rule to show that if f is homogeneous of degree n, then
x∂f
∂x + y∂f
∂y = nf (x, y).
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