M3220-1 Twelvth Homework Assignment

MATH 3220 § 1 TWELVTH HOMEWORK ASSIGNMENT Due Monday,

A. Treibergs November 26, 2007.

MATH 3220 § 1	TWELVTH HOMEWORK ASSIGNMENT	Due Monday,
A. Treibergs		November 26, 2007.

You are responsible for knowing how to solve the following exercises. Please hand in the starred "*" problems.

Please do the following exercises from the text "Foundations of Analysis" by Joseph L. Taylor.

10.1[ 2, 3, 4, 5*, 6*, 8, 9*, 10]

10.2[ 1, 2*, 3, 4*, 5*, 6, 7, 8, 12* ]

Please do the following additional exercises.

A*. Let γ : R -> R² be a continuously differentiable curve. Show that the image of a compact interval γ([a,b]) has Jordan content zero, called "volume zero" in the notes.
(Note that this would be false under the hypothesis of continuity only. This is because there exist "space filling curves." See, e.g., http://www.math.ohio-state.edu/~fiedorow/math655/Peano.html.)

You are responsible for knowing how to solve the following exercises. Please hand in the starred "*" problems.
Please do the following exercises from the text "Foundations of Analysis" by Joseph L. Taylor. 10.1[ 2, 3, 4, 5, 6, 8, 9, 10] 10.2[ 1, 2, 3, 4, 5, 6, 7, 8, 12* ] Please do the following additional exercises. A. Let γ : R -> R²* be a continuously differentiable curve. Show that the image of a compact interval γ([a,b]) has Jordan content zero, called "volume zero" in the notes. (Note that this would be false under the hypothesis of continuity only. This is because there exist "space filling curves." See, e.g., `http://www.math.ohio-state.edu/~fiedorow/math655/Peano.html`.)