(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. Prove that the functions ex, sin(x), cos(x) = F(R,R) are linearly independent.
E OFind y as a function of t if
6y" + 26y = 0,
y(0) = 9,
y(t) =
y' (0) = 4.
Note: This particular weBWork problem can't handle complex numbers, so write your answer in terms of sines and cosines,
rather than using e to a complex power.
Chapter 15 Solutions
Calculus Early Transcendentals, Binder Ready Version
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