Exercise 10.8

Question

Exercise 8: Let $H$ be the parallelogram in $\R2$ whose vertices are $(1,1)$, $(3,2)$, $(4,5)$, $(2,4)$. Find the affine map $T$ which sends $(0,0)$ to $(1,1)$, $(1,0)$ to $(3,2)$, $(0,1)$ to
    $(2,4)$. Show that $J_T=5$. Use $T$ to convert the integral
    $$
        \alpha=\int_He^{x-y}\,dx\,dy
    $$
    to an integral over $I^2$ and thus compute $\alpha$.

analambanomenos · Accepted Answer

Since $(3,2)=(1,1)+(2,1)$ and $(2,4)=(1,1)+(1,3)$, the linear part of the affine map is $A(u,v)=(2u+v,u+3v)$, so 
    \begin{align*}
        T(u,v) &= (1,1)+A(u,v)=(2u+v+1,u+3v+1) \
        J_T &=
            \begin{vmatrix}
                2 & 1 \
                1 & 3
            \end{vmatrix}
            =5 \
        \int_He^{x-y}\,dx\,dy &= \int_0^1\int_0^1e^{(2u+v+1)-(u+3v+1)}J_T\,du\,dv \
        &= 5\bigg(\int_0^1e^u\,du\bigg)\bigg(\int_0^1e^{-2v}\,dv\bigg) \
        &= 	extstyle\frac{5}{2}(e-e^{-1}+e^{-2}-1)
    \end{align*}

Exercise 10.8

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Exercise 10.8

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