If a course instructor closely follows a textbook when delivering a lecture (e.g. even listing the subsection they're discussing) could they be in violation of copyright law? What if they upload the lecture to YouTube? I'm particularly interested in the context of the United States.
I can't imagine an existing legal regime in which teaching from a textbook and "publishing" the lectures would be considered wrong unless the instructor explicitly made images or snippets copied from the book open to view. Likewise reading from the text verbatim would possibly be wrong depending on how much was read (fair use principles are discussed below). For a sentence or two, there are likely no issues. For a section, beware.
Referring the students to certain page/section numbers in a text is completely innocuous. Putting up page images is problematic.
Paging through a book with the camera pointed at the pages would probably be infringement everywhere, unless the snippets were very small (fair use). But, such paging seems to be blatant misuse in any case. Note, also, that the fair use rules for educational purposes are (normally) a bit relaxed from the general rules, though this has been changing.
But the lecture "belongs" to the lecturer, not the author/publisher of the book.
In some fields the standard for claiming a violation is much higher. If I copy the definition of the derivative (Calculus) onto a whiteboard, I'm not violating copyright even if my lecture is widely available. But that is because of a (typical) exception that things that can be said only one (or a very few) ways can't be copyrighted in the first place. So the author of Rudin (i.e. Rudin) used this exception in writing the book in the first place. But if I copy an equivalent amount from, say, a philosophy or history book, then there might be a problem.
You are much more likely to get a takedown notice at YouTube if you have some music playing in the background as you lecture. But that is because of draconian laws put in place by media publishers, not just copyright.
One of the (typical) standards for fair use is whether the use reduces the market for the item copied. If it does, then there are issues, otherwise there are not likely to be. If you "teach from" the book in such a way that students don't need access to it, then you should reexamine what you are doing. But even that implies that the "teaching from" is extensive. For supplementary materials, it might not be necessary for students to have other access. But there would be a "quantity" assessment made by anyone wanting to claim infringement and any court judging it. Does your use reduce the market?
And note that the laws are different everywhere, hence my use of "typical" etc. in a few places here. Perhaps a king could copyright the alphabet and make all written text subject to rules. See the book Fahrenheit 451 by Ray Bradbury for an extreme example.
I'll also note that, done right, such a course could actually increase the market for a textbook.
Oh hey a copyright hypo in real life :)
There are a couple questions that this question turns on.
Is the lecture a derivative work of the textbook? Generally, the holder of a copyright also has rights to derivative works. Derivative works are works which are derived from another work. For example, if I sell a t-shirt with a drawing of Mickey Mouse on it, that I made, Mickey Mouse is still under copyright and my drawing of him is a derivative work. It's not obvious to me that a lecture that merely references a textbook would count as a derivative work. You would need to adopt and apply the creative elements from the textbook to the work.
Is the material derived from the textbook a fact or a creative element? Generally, facts are not considered to be copyrightable (see e.g. Feist). An arrangement or selection of facts may be creative works subject to copyright, but that would also need to be shown. Furthermore, is the arrangement sufficiently distinguishable from other works (e.g. other textbooks)? Most Calculus textbooks start with differentiation and then discuss integration. Similarity between a lecture and a textbook merely on that basis would probably not be enough to establish the fact that the lecture is derivative.
The Digital Millennium Copyright Act (DMCA) contains several exemptions for instructors. See for example 17 USC Ch. 1 Section 110, where one is allowed to put on performances of certain copyrighted materials in an instructional context. You can watch a Mickey Mouse cartoon in class without obtaining rights to a public performance even though by any other metric it is a public performance. In this case, you're not performing exactly, but rather making a derivative work. I don't believe the right to derive is something that is ever explicitly contemplated in relation to instructors, but one could argue that in particular it is already implicit.
Lastly, to fall back on is fair use. The fair use exemption says that works that meet a weighing of the purpose of the derivative work, the nature of the copyrighted work, amount of copyrighted work used, and effect on the potential market for the copyrighted work will not be considered to infringe upon the copyright of the original work. Fair use falls more on the "standard" side of the rules-standard balance, so it's hard to say whether this claim would be successful. I would tend to say that it would be fairly likely given that the entire purpose of a textbook is to serve an instructional purpose, and presumably someone's buying the book if it's being used as a textbook. But at the same time, it's hard to say.
Of course, all of this is pretty much hypothetical. It doesn't address YouTube's copyright policy, nor does it really get around the fact that if a big textbook publisher wants to sue you, and you're just an average person, you probably won't have the resources to mount a legal defense.
I am not a copyright lawyer and I give no representations or warranty as to the accuracy of this legal opinion. If you choose to rely on this legal opinion, you do so entirely at your own risk. Please consult a copyright lawyer in the event that you need proper legal advice on this matter.
So far as I can see, the legal questions here come down to whether or not the activities you list fall within an exemption to the usual rules of copyright (e.g., "fair use" or other special exemptions). Copyright law in the US is governed under Title 17 of the United States Code. There are relevant exemptions in Sections 107 and 110. I am roughly familiar with the statutory framework here, but I am not a copyright specialist, and I am not familiar with the relevant case-law on these sections. Consequently, I can give you a rough idea of the law in this area, but you should consult a copyright lawyer with knowledge of relevant case-law if you want a more reliable answer.
Face-to-face lectures will generally be exempt from copyright
Section 110 gives a list of specific exemptions for a "performance or display of work", and subsection (1) includes displays of work for educational use. The full text of the relevant part of the section is shown below. So long as your teaching is occurring at a non-profit educational institution, and so long as you are using a legitimate copy of the textbook (i.e., a purchased copy, not a copy made in breach of copyright), the lectures themselves will fall within the scope of the exemption in §110(1).
§110 · Limitations on exclusive rights: Exemption of certain performances and displays
Notwithstanding the provisions of section 106, the following are not infringements of copyright:
(1) performance or display of a work by instructors or pupils in the course of face-to-face teaching activities of a nonprofit educational institution, in a classroom or similar place devoted to instruction, unless, in the case of a motion picture or other audiovisual work, the performance, or the display of individual images, is given by means of a copy that was not lawfully made under this title, and that the person responsible for the performance knew or had reason to believe was not lawfully made;
In the event that you are working for an educational institution that is a profit-based institution (i.e., not registered as a nonprofit) then this exemption will not apply, but you might be able to argue for an analogous exemption based on the general rules for "fair use" in §107. There is a reasonable case for fair use in an educational setting (even for a profit-based educational institution), especially since the textbooks are specifically written for this purpose. Indeed, one could reasonably argue that the textbook writers write these works in direct anticipation that they will be involved in a "display of work" in various educational settings.
Putting lectures on YouTube might be "fair use"
There is a general exemption under Section 107 for "fair use" of copyrighted works, which gives a range of factors that determine whether the exemption applies. This section is shown below.
§107 · Limitations on exclusive rights: Fair use
Notwithstanding the provisions of sections 106 and 106A, the fair use of a copyrighted work, including such use by reproduction in copies or phonorecords or by any other means specified by that section, for purposes such as criticism, comment, news reporting, teaching (including multiple copies for classroom use), scholarship, or research, is not an infringement of copyright. In determining whether the use made of a work in any particular case is a fair use the factors to be considered shall include—
(1) the purpose and character of the use, including whether such use is of a commercial nature or is for nonprofit educational purposes;
(2) the nature of the copyrighted work;
(3) the amount and substantiality of the portion used in relation to the copyrighted work as a whole; and
(4) the effect of the use upon the potential market for or value of the copyrighted work.
The fact that a work is unpublished shall not itself bar a finding of fair use if such finding is made upon consideration of all the above factors
These factors operate heavily in favour of an exemption in the situation you are proposing. Firstly, putting lectures on YouTube is something done for a nonprofit educational purpose. (If you are a user with a large audience then there may be some revenue from your posted videos on YouTube, but for most academic work that is likely to be miniscule in comparison to the time-cost to create the works. There could be some argument over whether the activity is really "nonprofit", but you will usually be on safe ground on that point.) Secondly, the nature of the copyrighted work is a textbook made for teaching purposes, so the broadcast of video lectures is consistent with this purpose. The amount and substantiality of the portion of the textbook used will depend on your lectures, but usually lectures consist of a substantial amount of supplementary discussion by the lecturer. Finally, to the extent that broadcasting your lecture on YouTube has an effect on the target market for the textbook, it is arguably likely to make that textbook more popular, rather than acting as a substitute good that would diminish sales.
My view is that, prima facie, these factors operate heavily in favour of providing an exemption. You would need to read some case-law on similar matters to see how courts view this, but it seems like you could make a good argument for an exemption with these factors. In view of this, my view is that you would probably be on safe ground uploading your lectures to YouTube, even if they closely follow the material in a copyrighted textbook.