An oath is a promise. If you want to borrow your brother's car, you may have to swear a solemn oath that you will bring it back unharmed.
An oath is a solemn promise, sometimes made in front of a witness, or a vow in a court of law that you will tell the absolute truth. No matter what the circumstances are, swearing an oath is serious business. That's why it's odd that an oath can also be an angry outburst of obscene words. Your brother might mutter an oath under his breath and clench his fists angrily if you return his car to him with a broken side mirror.