1. Let D ⊆ D ⊆ Rⁿ denote rectangles. Let f: D --> R. Assume F(x) ≥ 0, x ∈ D.
i) Assume ∫D₁ F does not exist. Show ∫D F does not exist.
ii) Assume ∫D F exists. Show ∫D₁ exists.

2. Let D = [0, 1] x [0, 1] ⊂ R² and assume f: D --> R is continuous. Define S ⊆ R³ by {(x,y,z) z = F(x,y), (x,y) ∈ D}. Show S has content 0.

3. Consider the transformation from R² --> R³ given by u = x² + y², v = xy. By the inversion theorem there exists an inverse transformation x = ψ(u,v) defined by F(u,v) in a neighborhood of (5,2) which maps (u,v) = (5,2) into (x,y) = (1,2).
i) Calculate ∂ψ/∂u(5,2).
ii) Find explicit functions x = ϕ(u,v), y = ψ(u,v)

4. Let D ⊆ Rⁿ be a rectangle. Let P, Q denote partitions of D. Let f: D --> R. Show L(P, F) ≤ u(Q, f).

5. Let Aₙₓₙ denote an invertible matrix with real entries. Define F: Rⁿ --> Rⁿ by F(x) = AX. Assume, a,b ∈ Rⁿ and F(a) = b, then the inversion theorem applies. Calculate the first two terms in the sequence, q1(x), q2(y) and show clearly q2(y) = q(y) = A⁻¹ y.

6. Let Aₙₓₙ denote a matrix with real entries. Assume A is positive definite and let λ be an eigenvalue of A. Prove that λ > 0.

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Functions and Transformations
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