1. Let Ω ⊂ R2 be the unit square: Ω = {x ∈ R² : 0 < x₁ < 1, 0 < x₂ < 1}. Define F: Ω → R² by

Verify that the divergence theorem holds for this domain Ω and this vector field F.

2. Define C_N²(Ω^-) = {u ∈ C²(Ω^-) : x ∈ ∂Ω ⇒ ∂u/∂n(x) = 0} and

L_N : C²_N(Ω^-) → C(Ω^-)

L_Nu = - Δu.

Show that LN is symmetric: (L_N u, v) = (u, L_N v)for all u, v ∈ C²_N(Ω^-).

3. Solve the BVP

-Δu = f(x) in Ω,

u = 0 on ∂Ω,

where Ω is the unit square in R² and f is the function.

4. Suppose Ω is the rectangle {x ∈ R² : 0 < x₁ < l₁, 0 < x₂ <  l₂}, and that u is a twice-continuously differentiable function defined on Ω^-. Let the Fourier sine series of u be

u(x₁, x₂) = _m=1Σ^∞ _n=1Σ^∞a_mnsin(mπx₁/l₁)sin(nπx₂/l₂).

where λ_mn = m²π²/l₁² + n²π²/l₂², m, n = 1, 2, 3, ...

5. Consider an iron bar, of diameter 4cm and length 1m, with specific heat c = 0.437 J/(g K), density ρ = 7.88 g/cm³, and thermal conductivity κ = 0.836 W/(cm K). Suppose that the bar is insulated except at the ends, it is heated to a constant temperature of 5 degree Celsius, and the ends are placed in an ice bath (0 degrees Celsius). Compute the temperature (accurate to 3 digits) at the midpoint of the bar after 20 minutes.