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<li>Instead of the functional notation for plugging vectors into | ||
differential forms, use the angle bracket pairing notation that is more | ||
standard and since we don't really use this much at all, there is no reason to | ||
have a less standard notation. This affects one place in chapter 1, | ||
exercise 4.4.3, and one place in appendix C. | ||
<li>The notation | ||
\(\C \otimes T_p M\) is changed to the more | ||
common \(\C T_p M\) which also avoids a slight abuse of notation. | ||
Thanks to Richard Lärkäng for the suggestion. | ||
<li>Lemma 2.3.9, the statement about the Levi form doesn't really make sense | ||
if \(M\) is not a boundary, so change the statement to explicitly say that | ||
the inside is "above" the \(M\) in the new coordinates. | ||
Also explicitly mention this at the end of the proof. | ||
Thanks to Richard Lärkäng for the suggestion. | ||
<li>At the beginning of section 2.4, give a reference to the one-variable | ||
book as not every student may have had a treatment of harmonic and | ||
subharmonic functions in a one-variable course. | ||
<li>Add hints to Exercise 2.4.8. | ||
<li>Change Proposition 2.4.5, to be about the sub-mean-value property | ||
for all small enough discs (which is really the best way to prove | ||
the original statement in the first place), | ||
leaving the original statement as an "In particular". | ||
That also | ||
<b>changes Exercise 2.4.11</b> by really making it a bit easier as it | ||
gives a way to do it (and change the hint to that exercise too). | ||
Thanks to Richard Lärkäng for the suggestion. | ||
<li>In Example 2.5.4, put a square on the norm of \(z\) as that makes the | ||
computation easier. | ||
Thanks to Richard Lärkäng for the suggestion. | ||
<li>In Theorem 2.5.6, add part (iv) which is just the conclusion of the | ||
second version of the Kontinuitatssatz as that just follows from the proof | ||
directly with no extra work. | ||
Thanks to Richard Lärkäng for the suggestion. | ||
<li>In Theorem 2.5.6, make part (ii) less wordy: "U is Hartogs | ||
pseudoconvex". I mean the definition is just above the theorem! | ||
<li>Mark Exercise 2.6.1 as Oka's lemma to make the connection to the "proof" | ||
above clear. | ||
<li><b>Change Exercise 4.4.1</b> to ask also to prove the Leibniz rule | ||
(which easily follows from the first part and the real Leibniz rule), but | ||
this is good to understand for the computations later in the section. | ||
Thanks to Richard Lärkäng for the suggestion. | ||
<li>Add Leibniz rule to Appendix C. | ||
<li>Simplify the wording of some of the theorems in Appendix E. | ||
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See https://www.jirka.org/scv/changes.html for older changes. |