Homework 1
Due: October 6, 2016
Problem 1. Prove ∫V
(ψ∇2ϕ−ϕ∇2ψ)dv =I(ψ
∇ϕ−ϕ
∇ψ)·ˆnda
Problem 2. Show that
A,
B,
Care not linearly independent if
A·(
B×
C) = 0
Problem 3. For any two scalar functions ϕand ψ, prove that
∇2(ϕψ) = ϕ∇2ψ+ψ∇2ϕ+ 2
∇ϕ·
∇ψ
Problem 4. If r is the vector from the origin to (x, y , z) prove that
∇· r = 3 ,
∇× r = 0 ,
(u ·
∇)r =u for any vector u
Problem 5. Derive the metric tensor and scaling factor of cylindrical coordinate
Problem 6. If
A= 2rˆar−3rsinϕ ˆaθ, find the representation of
Ain the cartesian coordi-
nate system
Problem 7. A vector c in cartesian coordinate system is c = 3 ˆax+ ˆay−3 ˆaz. Find its
representation using the following base vectors
ˆa1=1
√2( ˆax+ ˆay),ˆa2=1
√2( ˆax−ˆay),ˆa3= ˆaz
Problem 8. Evaluate ∫S
D·ds when
D=rsinθ ˆar+rsinθ ˆaθ, and Sis a unit sphere
centered at the origin
Problem 9. For g= 2xy +z2, find
a) the magnitude and direction of the maximum rate of change of gat point P(1,3,2)
b) the rate of change of galong the line directed from P(1,3,2) to P′(2,2,−1), evaluated
at P(1,3,2)
Problem 10. Find
∇ ×
Bif
B= 4rsinθ ˆar+ 3rcosϕ ˆaθ
Problem 11. Given f=rsinθcosϕ calculate
∇f,
∇ ×
∇fand
∇2f
Problem 12. Prove
∇2
A=
∇(
∇ ·
A)−
∇ × (
∇ ×
A) in cartesian coordinate
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