**July 23rd 2020 edition:**

- On page 57, $\frac{m-n}{2}$ should be $\max \{ \frac{m-n}{2} , 0 \}$
- Theorem 1.2.7 requires the hypothesis that $f^{-1}(0)$ has no limit points on $U' \times \partial D$.
- Mention that $\beta$ are the rows and $\alpha$ columns in section 2.3
- More explicitly give the computation in Example 2.3.1.
- Fix some typos in equation 2.20, the bars in second line should go all the way across and $\varphi(0) = p$ just above.
- Minor spelling, style, typo fixes.

**April 8th 2014 edition:**

- Update references, fix reference labels.
- Fix the Cauchy estimate inequality, it is only the point, not the norm on the left unless a smaller set would be taken.
- Slightly update the note in the beginning.

**February 28th 2011 edition:**

- Include local parametrization theorem for varieties without proof, and add exercise related to it.
- Fix statement of Theorem 1.2.7.
- Fix a number of typos found by Montgomery Taylor, thanks!
- Many minor grammar and typo fixes.

**December 2nd 2010 edition:**

- Include more detail in the proofs and improve exposition of the theorems in 2.4 and 2.6
- Add exercise in 1.4 about parametrization by discs which we actually use later.
- Minor clarifications.
- Many minor grammar and typo fixes.

**July 15th 2010 edition:**

- Fixed up hyperlinks, which are now underlined as well and have larger targets to click on.
- Added a bunch of references, more will come in the future sometime.
- Added some clarifications in places.
- Lots of spelling, grammar, and stylistic fixes.
- Using newest texlive so producing PDF 1.5.
- Using the microtype package with pdflatex to use microtypography. It really looks wonderful and pdflatex does all the work for you.

**May 11th 2010 edition:**

First version