In the glossary, Cowan defines a phrase as: a head and any modifiers (no gloss is supplied for head or modifier.) Consider a simple verb phrase such as (this grammar) bites the wax tadpole. On p. 20, we find that VPs consist of "a main verb, which is the head of the phrase... and any following NPs, PPs, AdjPs, or AdvPs that may be present." In our sentence, this leaves the wax tadpole as a modifier of bites, a particularly heterodox analysis. Auxiliary verbs are then not part of the VP (unless they head their own VPs, a point which doesn't seem to be addressed, but which would result in the bizarre notion that the main verb is a modifier of the auxiliary).
On pp. 18-19, TGE provides the following (apparently exhaustive) list of functions of NPs: subjects, direct objects, indirect objects, objects of a preposition, predicate nominals, and appositives. Presumably, then, NP itself is not considered a function, but rather a phrase. Yet on p. 16, gerunds are said to "function as NPs". Still later, on p. 472, they are said to be part (the head, I suppose) of gerund clauses, which, in turn, are said to function as subject clauses, and which are specifically contrasted with NPs on p. 471. So are they the same or are they not?
The list of NP functions is incomplete. In the sentence I started two days ago, the NP two days ago functions as an adjunct. Nouns can also function as determiners (e.g., Kawabata's short stories and modifiers (e.g., the faculty office).
On p. 22, it says that do expresses distinctions of tense and aspect. TGE recognises two aspects in which auxiliary verbs participate: perfect and progressive; do is not used for either.
In the chapter on imperative sentences, it states that the only inflected verb form allowed in an imperative after be is "in the progressive form". This ignores passives such as (e.g., be warned, be finished, be gone, don't be fooled by... etc.)
Page 113 has a section on "Idiomatic I need you Imperatives" (e.g., I need you to get ready quickly.) Such sentences are certainly hortative, but imperative is a grammatical term, not a rhetorical one. On p. 110, the main verb in an imperative is described as being "always in its bare infinitive form", yet I need you to..., clearly doesn't meet this requirement for inclusion. (Also note that imperatives do not have truth values, whereas this construction could be subject to the rejoinder, "no you don't.")
Overall, Cowan has simply not taken enough care to build a coherent grammatical system. It's not that I don't agree with his theories; it's that he doesn't agree with his own theories. Over and over again he contradicts himself or makes claims that are factually wrong.
I think the folks at Cambridge ESL need to get their editorial house in order too. This is the second grammar book they've published in the last few years that fails to meet the minimal standards for accuracy and consistency.
What a waste!