At 6:34 he computes the cross product between r_theta and r_z to see the orientation of that particular parametrization r_1 But what if he does r_z x r_theta? The result will be exactly in the opposite direction, as the cross product is anticomutative ( a x b = - ( b x a ) ). One can say that the product should keep the order the variables has when defining the parametrization, but choosing r(theta, z) vs r(z, theta) has not particular reason, right? In a case where the surface is more complicated, I'm not going to be sure which variable goes first, nor which way the cross product should be computed.
Cool! Way to go! Nice explanation
Excellent sir ...i like your explanation. From INDIA
At 6:34 he computes the cross product between r_theta and r_z to see the orientation of that particular parametrization r_1
But what if he does r_z x r_theta? The result will be exactly in the opposite direction, as the cross product is anticomutative ( a x b = - ( b x a ) ).
One can say that the product should keep the order the variables has when defining the parametrization, but choosing r(theta, z) vs r(z, theta) has not particular reason, right?
In a case where the surface is more complicated, I'm not going to be sure which variable goes first, nor which way the cross product should be computed.