Stokes and Divergence Theorems
1. Verify that Stokes theorem is valid for the vector field 𝐹⃑ (𝑥, 𝑦, 𝑧) = 〈𝑥𝑦, 𝑥𝑦, 𝑦𝑧〉 and 𝑆 being the surface of intersection
between the plane 𝑥 + 𝑦 + 𝑧 = 1 and the cylinder 𝑥 2 + 𝑦 2 = 9. Use the boundary of 𝑆 as the curve for the line integral.
2. Evaluate ∮𝐶 𝐹⃑ ∙ 𝑑𝑟⃑ where 𝐹⃑ (𝑥, 𝑦, 𝑧) = 〈0, 0, 𝑦𝑧〉 and 𝐶 is the curve of intersection of 𝑧 = 1 − 𝑥 2 − 𝑦 2 and the planes
𝑥 = 0, 𝑦 = 0, and 𝑧 = 0.
3. Evaluate the integral ∯𝑆 𝐹⃑ ∙ 𝑑𝑆⃑ for the vector field 𝐹⃑ (𝑥, 𝑦, 𝑧) = 〈𝑦, −2, 𝑥〉 and surface 𝑆 the rectangular box given by
|𝑥| ≤ 2, 0 ≤ 𝑦 ≤ 1, and 0 ≤ 𝑧 ≤ 1.
4. Evaluate the integral ∬𝑆 ∇ × 𝐹⃑ ∙ 𝑑𝑆⃑ for the vector field 𝐹⃑ (𝑥, 𝑦, 𝑧) = 〈𝑥 2 𝑧, 0, 0〉 and 𝑆 being the surface
𝑥 = 4 − 𝑦 2 − 𝑧 2 with 𝑥 ≥ 0.
Note: since any surface with the same boundary must be equivalent, you could choose to use a more simple surface
with the same boundary…
5. Faraday’s Law in electricity and magnetism states that
∮ 𝐸⃑⃑ ∙ 𝑑𝑟⃑ = −
𝐶
𝑑
⃑⃑ ∙ 𝑛̂ 𝑑𝐴
∬𝐵
𝑑𝑡
𝑆
⃑⃑
⃑⃑ × 𝐸⃑⃑ = − 𝜕𝐵.
Use stokes theorem to prove that this is equivalent to one of Maxwell’s Equations, namely ∇
𝜕𝑡
Stokes and Divergence Theorems
1. Verify that Stokes theorem is valid for the vector field 𝐹⃑ (𝑥, 𝑦, 𝑧) = 〈𝑥𝑦, 𝑥𝑦, 𝑦𝑧〉 and 𝑆 being the surface of intersection
between the plane 𝑥 + 𝑦 + 𝑧 = 1 and the cylinder 𝑥 2 + 𝑦 2 = 9. Use the boundary of 𝑆 as the curve for the line integral.
2. Evaluate ∮𝐶 𝐹⃑ ∙ 𝑑𝑟⃑ where 𝐹⃑ (𝑥, 𝑦, 𝑧) = 〈0, 0, 𝑦𝑧〉 and 𝐶 is the curve of intersection of 𝑧 = 1 − 𝑥 2 − 𝑦 2 and the planes
𝑥 = 0, 𝑦 = 0, and 𝑧 = 0.
3. Evaluate the integral ∯𝑆 𝐹⃑ ∙ 𝑑𝑆⃑ for the vector field 𝐹⃑ (𝑥, 𝑦, 𝑧) = 〈𝑦, −2, 𝑥〉 and surface 𝑆 the rectangular box given by
|𝑥| ≤ 2, 0 ≤ 𝑦 ≤ 1, and 0 ≤ 𝑧 ≤ 1.
4. Evaluate the integral ∬𝑆 ∇ × 𝐹⃑ ∙ 𝑑𝑆⃑ for the vector field 𝐹⃑ (𝑥, 𝑦, 𝑧) = 〈𝑥 2 𝑧, 0, 0〉 and 𝑆 being the surface
𝑥 = 4 − 𝑦 2 − 𝑧 2 with 𝑥 ≥ 0.
Note: since any surface with the same boundary must be equivalent, you could choose to use a more simple surface
with the same boundary…
5. Faraday’s Law in electricity and magnetism states that
∮ 𝐸⃑⃑ ∙ 𝑑𝑟⃑ = −
𝐶
𝑑
⃑⃑ ∙ 𝑛̂ 𝑑𝐴
∬𝐵
𝑑𝑡
𝑆
⃑⃑
𝜕𝐵
Use stokes theorem to prove that this is equivalent to one of Maxwell’s Equations, namely ⃑∇⃑ × 𝐸⃑⃑ = − 𝜕𝑡