Given a parametrization $r(u,v)$ of a surface, do we use $r_u\times r_v$ or $r_v\times r_u$ for the normal vector in a surface integral?

Question

I am confused a bit in surface integral . When we compute surface integral using parameterization of the surface then we first write the vector r in terms of parameters say u and v

Then we find the derivative of vector r with respect to u (let's call it r(u) )and then the derivative of vector r with respect to v (let's call it r(v) )

Now comes my confusion to get the unit normal vector we take the cross product of two differentiated vectors.

I am confused in which order shall I take the cross product shall I compute r(u) × r(v) or shall I compute r(v) × r(u) as both are not equal.

Is there a rule as to which order shall I apply as it makes my answers wrong many a times. Thanks

Answer 1

The order does not matter much. In any way you can take the cross product, the answer will differ by just a "-" sign. So, in one case, your answer will be positive and in the other case, negative. Usually we take the outward normal, which gives us the positive answer. If you wish, you can take the absolute value of the answer in the final step.