Section 9.16 Divergence theorem
Divergence theorem (also known as Gauss’ theorem):
𝑭 ∙ 𝒏 𝑑𝑆 =
𝑆
𝜵 ∙ 𝑭 𝑑𝑉
𝐷
D is a closed volume
S is the boundary of D
n represents normal vectors on S that point out of D
𝑭 ∙ 𝒏 𝑑𝑆 is the flux of F across dS
𝜵 ∙ 𝑭 is the divergence of F within D
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Section 9.16 Example 1
Verify the divergence theorem if:
𝐹 = 𝑥𝒊 + 𝑦𝒋 + 𝑧𝒌
D is bounded by:
𝑧 = 1 − 𝑥2 − 𝑦2 ; 𝑧 = 0
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Section 9.16 Example 2
Use Gauss’ divergence theorem to find:
𝑭 ∙ 𝒏 𝑑𝑆 if 𝐹 = −𝑦𝒊 + 𝑥𝒋 + 𝑧𝒌
𝑆
S is given by: 𝑥 2 + 𝑦 2 + 𝑧 2 = 4
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Section 9.16 Example 3
B is the surface of V. The value of the integral U
below is:
𝐹 = (𝑥 − 2𝑥𝑦𝑧)𝒊 + 𝑦 2 𝑧𝒋 + (3𝑧 + 𝑦)𝒌
𝑈=
𝐵
𝑦2
𝑭 ∙ 𝒏 𝑑𝑆
V: 𝑥 2 +
≤ 25
𝑧 ≥0
𝑧 ≤2
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Section 9.16 Application
Coulomb’s law:
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𝑄𝑞
𝑭=
𝒏
2
4𝜋𝜀𝑜 𝑟
Electric field:
𝑭
1
𝑄
𝑬= =
𝒏
2
𝑞 4𝜋𝜀𝑜 𝑟
Determine the flux of E through a sphere surrounding Q
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