His eldest son was Sadi Carnot followed in his father's footsteps as an army engineer, but his related achievements were not in math, but in physics, having described the Carnot cycle. He died of cholera at 36.
His younger brother Hippolyte Carnot fled France with their father after the defeat of Napoleon. After returning to France, he turned to philosophy and ultimately was made a senator-for-life.
His son Marie François was the fourth president of the Third French Republic. He was assassinated by an Italian anarchist.
His brother Marie-Adolphe was a mining engineer and chemist, for whom carnotite (a uranium ore) is named.
1: d'Alembert was the illegitimate son of the novelist, courtesan, ex-nun (and sister of a cardinal) Claudine Guérin de Tencin, and the chevalier Louis-Camus Destouches, an artillery commander. The infant was abandoned on the steps of a church in Paris. "The foundling was brought up by the wife of a glazier; in later years, when he became a celebrated mathematician d'Alembert spurned the overtures of his mother, preferring to be recognized as the son of his impoverished foster parents.
2: Although it follows quite simply from the rules of the game, I defy anyone to say that it's obvious that ii is a real number. Euler (naturally) pointed that out. e^(-pi/2), if you must know.