- #1

yungman

- 5,755

- 292

[tex]\frac{\partial (x,y,z)}{\partial (u,v,w)} \hbox { for 3-space and }\frac{\partial (x,y)}{\partial (u,v)} \hbox{ for 2-space }[/tex]

But for surface area of 3-space in the following example where the vector value function:

[tex]\vec {r} = u\hat{x} + u cos(v) \hat{y} + u sin(v) \hat{z} [/tex]

You can see this is like:

[tex]\frac{\partial (x,y,z)}{\partial (u,v)}[/tex]

Which I don’t see this from the book. My question is whether the Jacobian is still:

[tex]| \frac{\pratial \vec{r}}{\partial u} X \frac{\pratial \vec{r}}{\partial v}| [/tex] ?

This is the standard way of finding surface area of a 3-space object. But this is like transform from 2 space [tex](u,v)[/itex] to 3-space[itex](x,y,z)[/itex].