1. Let D âŠ† D âŠ† Râ¿ denote rectangles. Let f: D --> R. Assume F(x) â‰¥ 0, x âˆˆ D.
i) Assume âˆ«Dâ‚ F does not exist. Show âˆ«D F does not exist.
ii) Assume âˆ«D F exists. Show âˆ«Dâ‚ exists.

2. Let D = [0, 1] x [0, 1] âŠ‚ RÂ² and assume f: D --> R is continuous. Define S âŠ† RÂ³ by {(x,y,z) z = F(x,y), (x,y) âˆˆ D}. Show S has content 0.

3. Consider the transformation from RÂ² --> RÂ³ given by u = xÂ² + yÂ², v = xy. By the inversion theorem there exists an inverse transformation x = Ïˆ(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 âˆ‚Ïˆ/âˆ‚u(5,2).
ii) Find explicit functions x = Ï•(u,v), y = Ïˆ(u,v)

4. Let D âŠ† Râ¿ be a rectangle. Let P, Q denote partitions of D. Let f: D --> R. Show L(P, F) â‰¤ u(Q, f).

5. Let Aâ‚™â‚“â‚™ denote an invertible matrix with real entries. Define F: Râ¿ --> Râ¿ by F(x) = AX. Assume, a,b âˆˆ Râ¿ 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) = Aâ»Â¹ y.

6. Let Aâ‚™â‚“â‚™ denote a matrix with real entries. Assume A is positive definite and let Î» be an eigenvalue of A. Prove that Î» > 0.

1. Let D âŠ† D âŠ† Râ¿ denote rectangles. Let f: D --&gt; R. Assum...

1. Let D âŠ† D âŠ† Râ¿ denote rectangles. Let f: D --> R. Assum...