Originally Posted by

**gralla55** Thanks alot for your input! The reason I started wondering about this, was because I were never satisfied with the way my textbook explained the jacobian when introducing variable changes in multiple integrals.

Some of the stuff in your post I couldn't quite follow though. I've had an undergrad-level course in abstract algebra (and one in linear algebra), but I've never heard of k-linear alternating forms or of multilinear algebra. I will look it up.

But all right, back to the cross product. As you wrote, it is only a "formal determinant". It's absolute value should be:

1) the area of the parallellogram spanned by the two vectors in question.

The actual formula seems to compute:

2) the sum of the areas spanned by the vector components in the xy, xz and yz-planes, multiplied by a unit vector in the perpendicular direction in each case.

My question now is, is there an easy intuitive way to show why those two things are the same thing? Or is there only a messy non-intuitive way with pages of hairy algebra?

Thank you so much again!