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In English, the word "help" can be used for any of these cases to ask somebody
for help or to give a help to someone:

  * In a store, when a clerk (store worker) says: 

> Can I help you, sir?

  * With friends, when you see that your friend needs some help on math:

> Do you want a help with math?

  * When someone is carrying heavy things:

> Shall I help you?

  * When it's me who needs some help.

> I want a help to make cookies.

  * Help with household things (housework) such as cleaning, ironing:

> Son, please help me cleaning the dishes.

  * To thank:

> Thanks for your help.  
>  I'm happy that I could help you somehow.

In what situations are the Japanese expressions "手伝う", "手を貸す", and "助ける" (or
in the forms "手伝いましょうか?", "手伝ってくれ?", "手を貸して。", "助けてくれ。", :教えてあげる。", "教えよっか?")
used?
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2011-06-24T03:34:50.717
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words perspective
Can I help you?
I fully agree, this is much harder to translate (well) than one would expect.

There are hundreds of nuances and scenarios covered by the English "can I help
you", and you list a lot of them... So I'll focus on three very typical broad
categories (I'm sure people will give you more):

**Strangers:** Typically, offering your help finding directions to a tourist
who looks lost.

Rather than a direct "Can I help you?", any variations on "are you OK?"
(implying that you are ready to help) is probably the best way to go.
Depending on context, anything from a `大丈夫ですか？` to `道に迷っているんですか。`

**Close friends and family:** Where an equivalent of the informal "Need a
hand?" would be appropriate. Then `手伝う` can come handy... From a purely
cultural standpoint, I still would try to keep it sounding more like an offer
than a question. E.g. `手伝ってあげよう` ("let me help!") rather than (the otherwise
perfectly correct): `手伝って欲しい？` ("do you want me to help?").

Less close friends, subordinates or same-level colleagues would be variations
of the above (with appropriate use of polite verbal forms).

**Your boss** (or any person high-enough above you): is a different matter.
There are many sonkeigo-infused expressions to offer help. They all have in
common that you must make it sound like you are _asking for a favour_ , not
_doing_ them a favour.

One of my personal favourite sonkeigo expression for that is:

`お手伝いさせていただきます`

(with countless variants:)

`お手伝いさせていただけますか`

`お手伝いさせていただきましょうか`

`手伝わせていただけますか`

etc.

Which _literally_ means you are begging your boss to do you the favour of
accepting your help.

**Edit:** as for ways to thank somebody for their help. It is once again down
to context. Most basic (and literal) way would be: `手伝ってくれてありがとうございました` but if
it was a big favour/help, you can't go wrong by focusing on "the trouble
you've caused" (and apologising for it), in which case you'd use:
`迷惑をかけて申し訳無い` (for bigger favours) or a simple `すみません` (which means both
"sorry" **and** "thank you" in that context).
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2011-06-24T04:05:19.017
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Here is what I can think of now.

for store worker

> 何かお探しですか？

math,

> 数学に関して困った事あったら言ってくださいね

to help carrying heavy stuff

> お荷物お持ちいたしましょうか？

I wanna help to make cookies.

> クッキー作るとき手伝ってもいい？

Son, please help me cleaning the dishes.

> お皿洗ってくれる？

thanks for your help.

> 手伝ってくれてありがとうね / 助かった

It's my pleasure

> どういたしまして / よかった (for 助かった) よかった
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2011-06-24T04:24:31.347
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Use the context, Luke!

  * In a store, when a clerk (store worker) says : Can I help you, sir?  
"いらっしゃいませ！" with an inquiring gaze at the customer

  * With friends, when you see that your friend needs some help on math: Do you want a help with math? "大丈夫？助けてあげようか？"、"大丈夫？手伝ってあげようか？" (I allow "助ける" here, for math can put people in trouble :P)

  * Someone is carrying heavy things: Shall I help you?  
"ちょっと待って、手伝ってあげますよ。" with an inviting gaze before taking any action.  
"大丈夫ですか？手伝ってあげましょうか？"

  * It's me who needs some help. I want a help to make cookies.  
"お菓子作りたいんだけど…手伝ってくれる？"

  * Help with house things (housework): cleaning, ironing, .. Son, please help me cleaning the dishes.  
"○君、お母さんは忙しいから、皿洗いをしなさい。"

  * Thanks for your help.  
有難うございました。大変助かりました。すみません。

  * I'm happy that I could help you somehow.  
いいえ、いいえ、当然ですよ。
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2011-06-24T04:27:25.073
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