36
1 Mathematical Physics
+
4
0
y
2
8
+ y
ydy
4
+
y
2
8
− y
dy
(along y
2
= 8x)
= +
16
3
1.8 (a) It is sufficient to show that Curl F = 0
∇ × F =
i
j k
∂
∂ x
∂
∂ y
∂
∂z
2x y + z
2 x
2 2xz
= ˆ
i.0 − ˆ j(2z − 2z) + ˆ
k(2x − 2x) = 0
(b) dΦ = F. dr = ((2x y + z
2 ) ˆ
i + x
2 ˆ j + 2xz ˆ
k)).( ˆ
idx + ˆ jdy + ˆ
kdz)
= (2x y + z
2 ) dx + x
2 dy + 2xzdz
= (2x ydx + x
2 dy) + (z
2 dx + 2xzdz)
= d(x
2 y) + d(z
2 x) = d(x
2 y + xz
2 )
Therefore Φ = x
2 y + xz
2
+ constant
(c) Work done = Φ 2 − Φ 1 = 5.0
1.9 Let U = x + y; V = x − y
∂U
∂ x
= 1;
∂ V
∂ y
= −1
The curves y = x
2 and y
2
= 8x intersect at (0, 0) and (2, 4).
∂U
∂ x
−
∂ V
∂ x
dx dy =
S
(1 − (−1))dx dy = 2
2
x=0
2
√
2x
y=x 2
dxdy
= 2
2
0
2
√
2x
x 2
dy
dx = 2
2
0
(2
√
2
√
x − x
2 ) dx = 2
4
√
2
3
x
3/2 −
x
3
3
2
0
=
16
3
This is in agreement with the value obtained in Problem 1.7 for the line integral.
1.10 Use the divergence theorem
A . ds =
∇. A dν
But ∇. A =
∂
∂ x
x
3
+
∂
∂ y
y
3
+
∂
∂z
z
3
= 3x
2
+ 3y
2
+ 3z
2
= 3(x
2
+ y
2
+ z
2 ) = 3R
2
A . ds =
3R
2 dν =
(3R
2 )(4π R
2 dR)
= 12π
R
4 dR =
12
5
π R
5
1.11
c
A . dr =
(2y ˆ
i − 3x ˆ j + z ˆ
k).(dx ˆ
i + dy ˆ j + dz ˆ
k)
=
(2ydx − 3xdy + zdz)