This is one variant of the 3-d analog of the "shoelace formula" for area: https://en.wikipedia.org/wiki/Shoelace_formula#Generalization https://en.wikipedia.or

This is one variant of the 3-d analog of the "shoelace formula" for area: https://en.wikipedia.org/wiki/Shoelace_formula#Generalization https://en.wikipedia.org/wiki/Shoelace_formula#Generalizatio... The 2-d version is from the 18th century. I'd expect the 3-d version was probably known in the 19th century, but I haven't searched for a reference.