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微分積分

II b, c, d, e, f, g

補助演習問題

No. 12

1 6

つの平面

x = 0, y = 0, z = 0, x = 1, y = 1, z = 1

で囲まれる立方体の表面を

S

とする

.

ベクトル場

a(x, y, z) = xy i + y j + z

2

k

に対する次の面積分の値を求め よ

.

 ただし

, S

上の単位法線ベクトル

n

は外向きとする

.

∫∫

S

a · n dS.

2 a, h > 0

を定数とし

,

円柱領域：

x

2

+ y

2

≤ a

2

, 0 ≤ z ≤ h

の表面を

S

とおく

.

ベ クトル場

a(x, y, z) = (2x + y + 2z) i + (4y + x − 3z) j + ( − 5z + 2x − 3y) k

に対す る次の面積分の値を求めよ

.

 ただし

, S

上の単位法線ベクトル

n

は外向きとする

.

∫∫

S

a · n dS.

3

曲面

S : z = x

2

+ y

2

, z ≤ 4

を考え

,

ベクトル場

a(x, y, z) = xz i − 3yz j + z

2

k

に対する次の面積分の値を求めよ

.

 ただし

, S

上の単位法線ベクトル

n

は下向き

とする

. ∫∫

S

a · n dS.

参照

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