What is the ruling on a wife leaving her parents' home without her husband's knowledge, even though she informed him two days prior to her departure that she would be going out to buy some things?

If the wife seeks her husband's permission to leave her father's house, or if she predominantly believes he would approve of her going out to shop, then there is no blame upon her in that. However, if he does not permit her, and she does not predominantly believe he would approve, then it is not permissible for her to go out except out of necessity, because her going out in such a case would be considered an act of defiance and disobedience.