Gemara

(Beraisa): If a man told his wife 'I do not want all vows you will take' or 'this is not a vow', this has no effect;

(Beraisa): One may not say to his wife on Shabbos '(your vow is) annulled for you' or 'void for you' like he says during the week. Rather, he tells her 'take and eat (what she forbade)' or 'go drink', and the vow is void.

(R. Yochanan): He must be Mevatel the vow in his heart.

(Beraisa - Beis Shamai): On Shabbos, a man is Mevatel his wife's vow in his heart. During the week, he must say it;

Beis Hillel say, both of these are the same. It suffices to be Mevatel the vow in his heart. He need not say it with his lips.

(R. Yochanan): If a Chacham said 'your vow is annulled', or if a husband said 'your vow is permitted', this has no effect:

(Beraisa): "This is the Davar (word)" - a Chacham permits, a husband does not permit.

78b (Beraisa): Stringencies of affirmation over annulment are that silence affirms a vow, but does not annul it. If he thought in his heart to annul, it is not annulled.

Rishonim

The Rif brings the Gemara verbatim.

Rambam (Hilchos Nedarim 13:4): One who is Mevatel the vows of his wife or daughter need not say anything, and all the vows are Batel.

Rambam (6): If she vowed or swore not to eat or not to drink, and he said 'Mufar Leich', it is annulled, and she may eat or drink. If he gave to her food or drink and said 'take and eat or drink', she eats or drinks, and the vow is Batel automatically.

Rambam (7): One who annuls the vows of his daughter or wife must say the words with his lips. If he annulled in his heart, it is not annulled.

Radvaz: Beis Shamai and Beis Hillel argue only about whether or not one must verbalize Bitul during the week. This implies that all require to verbalize Hafarah.

Rambam (ibid.): One who is Mevatel need not say the words with his lips, rather, he is Mevatel in his heart and forces her to do (unlike her vow). Whether or not she did, the vow is Batel.

Lechem Mishneh: The Rambam came to answer the Ran's question, that Hafarah in his heart does not help. Telling her to eat shows that he does not want the vow; it is in place of explicit Hafarah. Perush ha'Mishnayos connotes that he must Mevatel in his heart only if she did not eat. The Ra'avad seems to be unlike Beis Hillel, who do not distinguish Shabbos from weekdays! Perhaps the Ra'avad holds like R. Eliezer mi'Mitz, who says that he says the Bitul inaudibly with his lips. Sometimes the Gemara calls this in his heart.

Kesef Mishneh: It seems that Hafarah works even if he did not annul in his heart. Even if he wanted the vow to endure, but he annulled so that she will not resent him, it is annulled.

Rambam (19): If one thought in his heart to annul, it is not annulled, like we explained. Therefore, if he annulled in his heart, he can still affirm.

Rosh (10:9): The Tana was imprecise. We hold like Chachamim, that Hafarah never works for the future! He should have said 'if a man said told his wife 'I do not want all that you vowed' to teach that an expression of Hafarah is required. Heter works retroactively, but Hafarah is from now and onwards, and without any reason. Therefore he must show that he intends for Hafarah. The Yerushalmi says that Beis Shamai and Beis Hillel agree that if he did not tell her 'take and eat', all agree that Hafarah in the heart does not work. They argue about when he said 'take and eat.' Beis Shamai say that this helps only on Shabbos. Beis Hillel say that it helps even during the week. R. Yochanan's teaching (he must annul the vow in his heart) is according to everyone. R. Eliezer mi'Mitz says that 'take and eat' is not Hafarah. If it were, why would R. Yochanan say that he must annul in his heart?! Surely, if he said 'take and eat', he intended to annul! Rather, (on Shabbos) he must say in his heart 'it is annulled for you', but he should not say this aloud with his lips.

Korban Nesan'el (3): R. Eliezer mi'Mitz holds that thinking the words is not enough. He must say them with his lips.

Ritva (DH Lo): Thinking in his heart is not enough, for this is Devarim sheb'Lev (unspoken intents). He must say them with his lips. The Gemara says that Bi'ur Chametz is in his heart, and one is Yotzei Birkas ha'Mazon b'Di'eved if he blessed in his heart, i.e. when he said these with his lips. If not, this is called 'Hirhur'. However, Hirhur suffices for Kiyum, for even silence helps.

Rosh (77b DH Omar): 'One may not say like he says during the week' suggests that he must say something during the week. Really, it is not needed even during the week. However, during the week one normally says the Hafarah so that his wife will know. The Beraisa forbids using the same expression like he uses during the week.

Question: If he must annul in his heart, why must he say 'take and eat'?

Answer (Rosh, ibid.): He says this so that she will know that he annulled. Alternatively, all agree that Hafarah in his heart does not work without 'take and eat.'

Ran (77b DH Echad): Some texts say 'after Shabbos, he must say (words of Hafarah) with his lips. Bitul in his heart helps only for Shabbos. This is unreasonable. Beis Hillel say 'both of these (Shabbos and weekdays) are the same! Rather, the Gemara asked because telling her to eat is no better than saying 'it is not a vow' or 'I do not want you to vow.' Telling her to eat helps only when he also annulled in his heart.

Ran (79a DH Hefer): Silence for a day affirms, because he shows that he wants the vow. This shows that Kiyum in the heart suffices. He has a day to annul, for until then it is not clear that he wants Kiyum. Hafarah in his heart does not help only when he did not say 'take and eat.'

Poskim

Shulchan Aruch (YD 234:38): If one thought the words of Hafarah in his heart, but did not say them with his lips, nothing happened. This is when he did not tell her 'take and eat.' If he told her 'take and eat' and thought the words of Hafarah in his heart, this is Hafarah.

Beis Yosef (DH Chishev): The R. Eliezer mi'Mitz holds that R. Yochanan said that he must annul in his heart, i.e. he need not say the words audibly, but he must say them with his lips. The Tur holds that R. Yochanan connotes otherwise (that he need not even say them with his lips).

Shulchan Aruch (41): If one thought in his heart to annul, it is not annulled. Therefore, if he annulled in his heart, he can still affirm.