Chap1-HW1.pdf

Assignment/Homework (Chapter 1: PHYS 311) Due date: 5 Sept 2024 (during class) Total point: 35% "⃗ = 2𝑥𝑧 𝑥) + (𝑥 + 2)𝑦) + 𝑦(𝑧 ! − 3)𝑧̂ over the five sides 1. Calculate the surface integral of 𝑉 (e...

Assignment/Homework (Chapter 1: PHYS 311) Due date: 5 Sept 2024 (during class) Total point: 35% "⃗ = 2𝑥𝑧 𝑥) + (𝑥 + 2)𝑦) + 𝑦(𝑧 ! − 3)𝑧̂ over the five sides 1. Calculate the surface integral of 𝑉 (excluding the bottom) of the cubical box. Let “upward and outward” be positive direction as indicated by the arrows. (Hint: taking one side at a time) (10%) "⃗ = 𝑦 ! 𝑥) + (2𝑥𝑦 + 𝑧 ! )𝑦) + (2𝑦𝑧)𝑧̂ and a 2. Check the divergence theorem using the function 𝑉 unit cube at the origin as shown. (10%)  "! $! &! 3. The equation giving a family of ellipsoids is: 𝑢 = #! + %! + ' !. Find the unit vector normal to each point of the surface of these ellipsoids. (Hint: find the unit vector 𝑛)) (5%) 4. Given the vector 𝐴⃗ = 4𝑟̂ + 3𝜃7 − 2𝜑), find its line integral around the closed path shown in Figure shown. The curved portion is the arc of a circle of radius r0 centered at the origin. Also find the surface integral of ∇ × 𝐴⃗ over the enclosed area and compared your results. (10%)