Can the Jacobian be used for out of plane rectangles?

[TheFerruccio]

Suppose I wanted to know the surface integral of a recently whose points are (0,0,0),(0,0,2),(1,1,0),(1,1,2)

The integral itself, if the surface is parameterized in terms of u and v, would be in those two variables, a differential element whose sides are du and dv.

However, since this rectangle requires 3 variables to describe it, how would the Jacobian be constructed? Could that be used for the coordinate transformation? (suppose the integral boundaries are 0 and 1 for both u and v)

Thanks in advance for the help.

 

[LCKurtz]

Suppose I wanted to know the surface integral of a recently whose points are (0,0,0),(0,0,2),(1,1,0),(1,1,2)

The integral itself, if the surface is parameterized in terms of u and v, would be in those two variables, a differential element whose sides are du and dv.

However, since this rectangle requires 3 variables to describe it, how would the Jacobian be constructed? Could that be used for the coordinate transformation? (suppose the integral boundaries are 0 and 1 for both u and v)

Thanks in advance for the help.

You could let u be the vector from (0,0,0) to (1,1,0) and v be the vector from (0,0,0) to (0,0,2) so

$$\vec u = \langle 1,1,0\rangle,\, \vec v = \langle 0,0,2\rangle$$

and parameterize the rectangle in terms of these two vectors:

$$\vec R(s,t) = s\vec u + t\vec v = \langle s,s,2t\rangle,\, 0 \le s,t \le 1$$

Then use the standard formula for dS for a parameterized surface:

$$dS = |\vec R_s \times \vec R_t|ds\,dt$$

and calculate the area by

$$\int_0^1\int_0^1 1\, dS$$