My wife works too, yet I feel that cooking, cleaning, and the children are simply her department. Is that division actually how Islam sees it?

Brother, this is a fair question, and the honest answer is that the division you describe is not what Islam laid down. You provide, you carry burdens outside the home, and none of that is small. But if we want to know how the work of a house is meant to be shared, the place to look is the house of the Prophet, peace be upon him.

Aisha, may Allah be pleased with her, was once asked what the Prophet did inside his home. She did not describe a man who considered the household beneath him. She said plainly:

“He was in the service of his family, and when the time for prayer came, he would go out to pray.”

Aisha (may Allah be pleased with her), Sahih al-Bukhari (676) · meaning

The most honoured man who ever walked the earth mended his own sandal, helped with the chores, and served those under his roof, and it took nothing from his dignity. If the hands that received revelation could serve the home, our hands are not lowered when they do the same.

The idea that the kitchen and the children are one person’s “department” is a custom we inherited from our surroundings, a habit of the times, not a ruling of our faith. And of the believers as a whole, in a verse whose own theme is how the faithful stand together, calling to what is good, holding up the prayer and giving in charity, Allah says:

“And the believing men and the believing women are protectors and helpers of one another.”

Qur’an, At-Tawbah 9:71 · meaning

Awliya of one another: a shared shelter and a shared load. Let me be open with you about how I am reading this verse. It speaks of the believing community at large, not of husband and wife in particular, so the lesson for your home is mine to draw, and I draw it without hesitation. If this is how Allah describes believers who meet one another in the street and the masjid, what shall we say of two believers who share one roof, one table and one set of children? That mutual care does not stop at the door of the home where it is needed most. When your wife carries a paid day as you do and then a second unseen shift at home, her strength quietly wears thin. Even those who study homes today notice that when the load feels fair to both, a husband and wife grow closer, while a share that one of them quietly feels is unfair slowly tires the heart. Our faith reached that truth long before any study, through the example of a Prophet who picked up a cloth and helped.

So think of it not as her department and yours, but as one home you are both building toward Jannah, where every dish you wash and every child you settle becomes, by your intention, an act that Allah records and rewards.

One small step: Pick one household task this week, the dishes after dinner or putting the children to bed, and quietly make it yours, following the one who served his own family.