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I am new to this forum and even don't know how to write these mathematical symbols. I have a question below and your reply would be highly appreciated.

If I have a triangle with vertices $V_1 = (2,2)$, $V_2 = (4,2)$, $V_3 = (2,4)$. I can find its area by integrating as follows: $$\int_2^4 \int_2^{6-x} dydx \text{(this gives area 2)}.$$

Now if I transform $V_1 = (2,2)$ to $U_1 = (0,0)$ and the vertices $V_2$ and $V_3$ to a standard simplex using relation $$ I = TS $$ where $I$ in identity matrix (the standard simplex), $T$ is the transformation matrix and $S$ is the 1-simplex defined by $V_2$ and $V_3$. It means I used transformation matrix $T$ to transform $V_2$ and $V_3$ to new vertices $U2 = (1,0)$ and $U3 = (0,1)$ respectively (transformation from non-standard simplex $(V_1,V_2)$ to a standard simplex $(U_1,U_2)$).

How can I use the effect of this transformation matrix $T$ in my new integral to get the same area as that of the original triangle with vertices $V_1$, $V_2$, and $V_3$?

Orat
  • 4,065
assadabbasi
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  • You need to form the Jacobian of that transformation and integrate that over your new simplex. – Ron Gordon Mar 22 '13 at 10:42
  • @RonGordon thanks for your reply. I have solved the problem using change of variables i.e. x = 2u+2 and y = 2v+2. I used partial derivatives of x and y with respect to u, v to form the Jacobian matrix and then taking the determinant, i used that value in new integral corresponding to U1, U2 and U3. So got the correct answer. But if i use the transformation I=TS, how will i use this 'T' to form the Jacobian? T is a matrix T = [0.3333 -0.1667; -0.1667 0.3333]. And in this case change of variables will remain the same as I described above? – assadabbasi Mar 25 '13 at 06:50
  • Welcome to MSE! For mathematical typing, see http://meta.math.stackexchange.com/questions/107/faq-for-math-stackexchange/117#117 – Orat Mar 28 '13 at 05:45

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