Learn Hardcore Japanese: atui hi ni kawa de oyogimasen ka? - Shall we swim in the river on a hot day?

Because で indicates the location of an action.
• 川で泳ぐ = “swim in the river.”
In contrast, に after a place often marks a destination (e.g., 川に行く “go to the river”) or a static location.

Why is the invitation phrased as 泳ぎませんか (negative question) rather than 泳ぎましょうか?

The negative‐question form ～ませんか is a standard polite way to invite someone, softening the suggestion.
• 泳ぎませんか → “Won’t you swim with me?”/“Shall we swim?”
• 泳ぎましょうか can sound like “Shall I swim?” or “Shall I do the swimming for you?” rather than a joint invitation.

What happens if we drop the final か from 泳ぎませんか?

Without か, 泳ぎません simply becomes a plain negative statement—“(I/We) won’t swim.”
The か at the end turns it into a negative‐question (i.e., an invitation).

Why is there no explicit subject in this sentence?

Japanese often omits the subject when it’s clear from context.
Here, it’s implicitly “you and I” or “us,” so you don’t need to say 私たちは or あなたは.

Could we use は instead of に after 暑い日?

Yes. 暑い日は川で泳ぎませんか is also correct.
• に simply marks the time point (“on a hot day”).
• は turns 暑い日 into a topic (“As for hot days…”), adding a slight nuance of general habit or contrast.

How do you form similar polite invitations with other verbs?

Use the verb’s ます‐stem + ませんか.
Examples:
• 食べませんか？ “Won’t you eat (with me)?”
• 行きませんか？ “Shall we go?”
• 見ませんか？ “Want to watch/see (something)?”

Why are there spaces between 暑い, 日, and に in the example?

Those spaces are just for teaching/segmenting purposes. In real Japanese text, you wouldn’t space them out.
Correct continuous writing:
暑い日に川で泳ぎませんか？