Is there a transformation whose differential is the following

Question

Let $T$ be a transformation

is $$dT = \begin{bmatrix} y & x \\ xy & x+y\end{bmatrix}$$

Any hints?

Answer 1

Hint:

If you are searching a function $$ T:\mathbb{R}^2 \to \mathbb{R}^2 \qquad T(x,y)=(T_1(x,y),T_2(x,y)) $$ such that $$ dT = \begin{bmatrix} \frac{\partial T_1}{\partial x} & \frac{\partial T_1}{\partial y} \\ \frac{\partial T_2}{\partial x} & \frac{\partial T_2}{\partial y}\end{bmatrix}= \begin{bmatrix} y & x \\ xy & x+y\end{bmatrix} $$

than note that from the second row, we have: $$ T_2(x)=\int xy dx =\frac{1}{2}x^2y+C(y) $$ but $$ \frac{\partial }{\partial y} \left(\frac{1}{2}x^2y+C(y) \right) $$ cannot be $x+y$