change of coordinates

$ = \frac{\partial y^k}{\partial x^i} \left( \frac{\partial u^j}{\partial x^m} \frac{\partial x^m}{\partial y^k} \frac{\partial y^l}{\partial x^j} \frac{\partial}{\partial y^l} + u^j \big(\vec{x}(\vec{y})\big) \frac{\partial^2 y^l}{\partial y^k \partial x^j} \frac{\partial}{\partial y^l} \right)

$ = \frac{\partial y^k}{\partial x^i} \left( \frac{\partial u^j}{\partial x^m} \frac{\partial x^m}{\partial y^k} \frac{\partial y^l}{\partial x^j} \frac{\partial}{\partial y^l} \right) + \frac{\partial y^k}{\partial x^i} \left( u^j \big(\vec{x}(\vec{y})\big) \frac{\partial^2 y^l}{\partial y^k \partial x^j} \frac{\partial}{\partial y^l} \right)

$ = \left( \frac{\partial u^j}{\partial x^i} \frac{\partial y^l}{\partial x^j} \right) \frac{\partial}{\partial y^l} + \frac{\partial y^k}{\partial x^i} \left( u^j \big(\vec{x}(\vec{y})\big) \frac{\partial^2 y^l}{\partial y^k \partial x^j} \frac{\partial}{\partial y^l} \right)

$ = \left( \frac{\partial u^j}{\partial x^i} \frac{\partial y^l}{\partial x^j} + u^j \big(\vec{x}(\vec{y})\big) \frac{\partial y^k}{\partial x^i} \frac{\partial^2 y^l}{\partial y^k \partial x^j} \right) \frac{\partial}{\partial y^l}