# Surface integrals in R^n, n>3

1. Nov 9, 2009

### logarithmic

How would I calcluate a surface integral in dimensions greater than 3.

For example, from the definition of a surfrace integral over a vector field: http://en.wikipedia.org/wiki/Surface_integral#Surface_integrals_of_vector_fields

To compute the surface integral, I would first need a vector normal to the vector field. In R^3 this is just done by using the cross product. Is there a general way to find a normal vector when not in R^3, since the cross product is no longer valid?

2. Nov 9, 2009

### Hurkyl

Staff Emeritus
Actually, what you need is the a bivector tangent to the surface.

That trick works in R3 because bivectors can be identified with "axial vectors". However, in R4, the dual would be some sort of "axial bivector", and in R5 it would be an "axial trivector" -- so we can't use this trick anymore.