Is it rude to request a book chapter from the academic who wrote it if you don't want to buy the entire textbook? The title says it all. I want to read a book chapter from a prominent Oxbridge researcher, but I don't want to shell out the cash for the entire textbook in which it's published. Is it rude to contact the professor who wrote the chapter and request an electronic version? It's for my own research purposes.
Before writing to the author, you should try to find the chapter in other ways:


Is it available online?  You should make sure it's not in any obvious place (e.g., the author's home page or university web site) or findable by a web search.
Do you have library access?  Even if the library doesn't have the book, they can very likely get it through interlibrary loan.
Do you have friends, colleagues, or teachers who might have the book?


It's a little rude to bother the author to ask for a copy if you could reasonably get one another way.  However, if the book is unaffordable for you and you have no other options, then there's nothing wrong with asking the author.  I'd phrase it as a question, to avoid sounding too demanding.  The key thing to keep in mind is that you're asking for something unusual, just in case the author has an electronic copy they'd be willing to share (but haven't put on their web site).  It's also worth including a sentence about things like your lack of library access.  You may not get a copy, but it's worth a try.