Is expiation (kaffarah) obligatory for a woman who swore not to go anywhere but to one specific place, then went to another after her husband made it permissible for her?

The default is that a breach of an oath occurs by going to a second location, and expiation for the oath becomes obligatory, unless you intended, at the time of your oath, that you would not enter the second location except with your brother's consent. In that case, you do not breach the oath and expiation is not required of you. Similarly, if the motive for the oath was your brother's disapproval of you entering a second location, then you do not breach the oath if he consents to your entry; because the intention of the swearer and the reason that prompted the oath are considered in restricting and specifying the wording of the oath.