Question:
Is a wife who is financially independent (i.e. her husband does not support her) allowed to work during the week that her husband is sitting shiva.
Answer:
Under normal circumstances the wife and children still at home cannot work if the husband is sitting shivah. The rwason being is that their earnings belong to the husband/father. But if the husband tells his wife explicitly that her wages belong only to her then she is allowed to work.
Sources:
Y:D 380:4, Daas Kedoshim ibid.