(a) In Example 3 of Section 15.1 we showed that ϕ x , y = − c x 2 + y 2 1 / 2 is a potential function for the two-dimensional inverse-square filed F x , y = c x 2 + y 2 3 / 2 x i + y j but we did not explain how the potential function ϕ x , y was obtained. Use Theorem 15.3.3 to show that the two-dimensional inverse-square filed is conservative everywhere except at the origin, and then use the method of Example 4 to derive the formula for ϕ x , y . (b) Use an appropriate generalization of the method of Example 4 to derive the potential function ϕ x , y , z = − c x 2 + y 2 + z 2 1 / 2 for the three-dimensional inverse-square filed given by Formula (5) of Section 15.1.
(a) In Example 3 of Section 15.1 we showed that ϕ x , y = − c x 2 + y 2 1 / 2 is a potential function for the two-dimensional inverse-square filed F x , y = c x 2 + y 2 3 / 2 x i + y j but we did not explain how the potential function ϕ x , y was obtained. Use Theorem 15.3.3 to show that the two-dimensional inverse-square filed is conservative everywhere except at the origin, and then use the method of Example 4 to derive the formula for ϕ x , y . (b) Use an appropriate generalization of the method of Example 4 to derive the potential function ϕ x , y , z = − c x 2 + y 2 + z 2 1 / 2 for the three-dimensional inverse-square filed given by Formula (5) of Section 15.1.
(a) In Example 3 of Section 15.1 we showed that
ϕ
x
,
y
=
−
c
x
2
+
y
2
1
/
2
is a potential function for the two-dimensional inverse-square filed
F
x
,
y
=
c
x
2
+
y
2
3
/
2
x
i
+
y
j
but we did not explain how the potential function
ϕ
x
,
y
was obtained. Use Theorem 15.3.3 to show that the two-dimensional inverse-square filed is conservative everywhere except at the origin, and then use the method of Example 4 to derive the formula for
ϕ
x
,
y
.
(b) Use an appropriate generalization of the method of Example 4 to derive the potential function
ϕ
x
,
y
,
z
=
−
c
x
2
+
y
2
+
z
2
1
/
2
for the three-dimensional inverse-square filed given by Formula (5) of Section 15.1.
5) Determine for which value of m the function
$(x) = x" is a solution to the given equation.
(a) 3,2d³y
dr?
dy
+ 11x
3y = 0
dx
dy
(b)
dy
- 5y = 0
dx
dx2
If Ø(z) = y + ja represents the complex potential for an electric field and
α = 2² +
+ (x+y)(x−y), determine the function Ø (z)?
(x+y)²-2xy
Precalculus: Mathematics for Calculus (Standalone Book)
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