(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.
8. Consider the function
y = c₁e² cos 2x + c₂e sin 2x + c3 + C₁x + c5€¯ + c6е
2x
(a) r²(r - 1)(r-2) (r + 1) (r + 2)² = 0
(b) r²(r 1)²(r - 2)(r + 1)² (r + 2)² = 0
(c) r²(r− 1)² (r² + 4)(r + 1)(r + 2)² = 0
(d) r(r² − 2r+5)(r + 1)(r + 2)² = 0
(e) r²(r² - 2r+5)(r+1)(r+ 2)² = 0 ✓
+ C7xe
-2x
with the c, are arbitrary constants. This y is the general solution to a certain homogeneous linear
first-order ODE. Which of the following is its auxiliary equation?
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
Chapter 15 Solutions
Calculus Early Transcendentals, Binder Ready Version
Precalculus: Mathematics for Calculus (Standalone Book)
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