HW 3 Suggested Problems (you do not have to turn them in).
(1) In section 14.6: do exercise: 8, 10, 12, 16, 17 (2) In section 16.5: do exercise: 5, 6, 7
Turn in the following set of problems:
(1) In section 14.6: 37
(2) In section 16.5: 23, 24, 25, 26, 27, 28, 29, 30.
(3) Letz=f(x, y) be a nice function defined on a domainDonR2.Letu=iandv= (i+j)/√ 2.
Suppose thatpis an interior point ofD andDuf(p) = 3 andDvf(p) =√ 2.
(a) Find∇f(p).
(b) Find the maximum ofDwf(p) whenw runs through the set of unit vectorsS1={w∈ R2:kwk= 1}.
(c) Find allwsuch thatDwf(p) = 0.
(4) Let u : D → R be a nice function on a plane domain D such that uxy = 0 on D. Find constantsa, bsuch that the functionf :D→Rdefined byf(x, y) =u(x, y)eax+bysatisfying the following partial differential equation
fxy−2fx−3fy+abf= 0.
(5) Let k, Q, qbe real numbers. We define a function V :R3\ {0} →Rby V(x, y, z) = kQq
r wherer=p
x2+y2+z2. (a) Find∇V.
(b) LetC be a curve from (2,2,1) to (3,4,12). Find the line integral Z
C
∇V ·dr.
(c) Prove that ∆V = 0 where ∆ =∇ · ∇is the Laplace operator.
(6) Let Fbe a nice vector field on a regionD in R3.Prove that∇ ·(∇ ×F) = 0.
(7) Let u:R2\ {0} →Rbe a nice function. We define a functionf : (0,∞)×[0,2π)→Rby f(r, θ) =u(rcosθ, rsinθ).
Use chain rule to prove that
uxx+uyy=frr+1 rfr+ 1
r2fθθ. (This is the Laplacian in the polar coordinate).
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