(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.)
Consider the line perpendicular to the surface z =
x2 + y at the point where x =
1 and
y =
-1.
Find a vector parametric equation for this line in terms of the parameter t. Enter your answer using vector notation .
L(t) =
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