SRM Institute of Science and Technology
Department of Mathematics
21MAB102T-Advanced Calculus and Complex Analysis
2022-2023 Even
Answer all questions (5 X 10 = 50 marks)
Assignment - 1
S.No.
1. Find the area inside the cardioid r = a(1+cos𝜃) and outside the circle 𝑟 = 𝑎 .
2. Evaluate ∭ 𝑥𝑦𝑧𝑑𝑥𝑑𝑦𝑑𝑧, where V is the volume given in the positive octant of the sphere
𝑉
𝑦2 + 𝑧2 = 1 .
𝑥2 +
3. Show that 𝐹̅ =(𝑧 2 + 2𝑥 + 3𝑦)𝑖̂ + (3𝑥 + 2𝑦 + 𝑧)𝑗̂ + (𝑦 + 2𝑧𝑥 )𝑘̂ is irrotational but not solenoidal.
Hence find its scalar potential.
4. Verify Green’s theorem in a plane with respect to ∫ 𝑥 2 𝑑𝑥 − 𝑥𝑦𝑑𝑦 , where C is the boundary of the
𝐶
square formed by 𝑥 = 0, 𝑦 = 0, 𝑥 = 𝑎, 𝑦 = 𝑎 .
5. Verify the Stoke’s theorem for 𝐹̅ = (𝑦 − 𝑧 + 2)𝑖̂ − (𝑦𝑧 + 4)𝑗̂ − (𝑥𝑧)𝑘̂ over the surface of a cube
𝑥 = 0, 𝑦 = 0, 𝑧 = 0, 𝑥 = 2, 𝑦 = 2, 𝑧 = 2 above the XOY plane.