(a) Show that when Laplace’s equation ∂ 2 u ∂ x 2 + ∂ 2 u ∂ y 2 + ∂ 2 u ∂ z 2 = 0 is written in cylindrical coordinates, it becomes ∂ 2 u ∂ r 2 + 1 r ∂ u ∂ r + 1 r 2 ∂ 2 u ∂ θ 2 + ∂ 2 u ∂ z 2 = 0 (b) Show that when Laplace’s equation is written in spherical coordinates, it becomes ∂ 2 u ∂ ρ 2 + 2 ρ ∂ u ∂ ρ + cot ϕ ρ 2 ∂ u ∂ ϕ + 1 ρ 2 sin 2 ϕ ∂ 2 u ∂ θ 2 = 0
(a) Show that when Laplace’s equation ∂ 2 u ∂ x 2 + ∂ 2 u ∂ y 2 + ∂ 2 u ∂ z 2 = 0 is written in cylindrical coordinates, it becomes ∂ 2 u ∂ r 2 + 1 r ∂ u ∂ r + 1 r 2 ∂ 2 u ∂ θ 2 + ∂ 2 u ∂ z 2 = 0 (b) Show that when Laplace’s equation is written in spherical coordinates, it becomes ∂ 2 u ∂ ρ 2 + 2 ρ ∂ u ∂ ρ + cot ϕ ρ 2 ∂ u ∂ ϕ + 1 ρ 2 sin 2 ϕ ∂ 2 u ∂ θ 2 = 0
Solution Summary: The author explains the Laplace equation in cylindrical coordinates. The spherical coordinate is (r,theta,z).
(a) Show that when Laplace’s equation
∂
2
u
∂
x
2
+
∂
2
u
∂
y
2
+
∂
2
u
∂
z
2
=
0
is written in cylindrical coordinates, it becomes
∂
2
u
∂
r
2
+
1
r
∂
u
∂
r
+
1
r
2
∂
2
u
∂
θ
2
+
∂
2
u
∂
z
2
=
0
(b) Show that when Laplace’s equation is written in spherical coordinates, it becomes
∂
2
u
∂
ρ
2
+
2
ρ
∂
u
∂
ρ
+
cot
ϕ
ρ
2
∂
u
∂
ϕ
+
1
ρ
2
sin
2
ϕ
∂
2
u
∂
θ
2
=
0
Represent the line segment from P to Q by a vector-valued function. (P corresponds to t = 0. Q corresponds to t = 1.)
P(0, 0, 0), Q(2, 7, 5)
r(t) =
Represent the line segment from P to Q by a set of parametric equations. (Enter your answers as a comma-separated list.)
Represent the line segment from P to Q by a vector-valued function. (P corresponds to t = 0. Q corresponds to t = 1.)
P(0, 0, 0), Q(4, 5, 4)
r(t) =
Represent the line segment from P to Q by a set of parametric equations. (Enter your answers as a comma-separated list of equations.)
8. A line initiated from (1,0,2) and parallel to a vector i+j+k what is the parametric equation of the line
(if t is a parameter)?
A. x=1+ 2i, y= 2j , z= 2+j
B. x=1+ 2t, y= 2t
C. x=1+ t, y=t, z= 2+t
D. x=2+ i, y= 1,
z= 2+t
9. Find , dA, where r is a radius of a circle
1e30 Jr=0
В. п/2 С. 4л D. т/4
А. 2л
10. Which of the following is not a function of f: R? → R?
A. f(r, y) = ey
B. f(r, y) = 1 C. f(r, y) = e +1
D. f(r) = e"
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