# Show that vector field is $C^\infty$ and calculate its Jacobian matrix at $(0,1,1)$

From an introductory course on calculus on manifolds:

Show that $$f : U \subset \Bbb R^3 \to \Bbb R^2$$ given by $$f(x,y,z) = \begin{bmatrix} \sin(x+z) + \log(y z^2) \\ e^{x+z} + y z \end{bmatrix}$$ is $$C^\infty$$ and calculate its Jacobian matrix at $$(0,1,1)$$ where $$U = \left\{ (x, y, z) \in \Bbb R^3 \mid y, z > 0 \right\}$$

I already showed that $$f$$ is continuous and that it is $$C^1$$, but I don't really get how to use induction from here. Any help?

• A composition of $C^\infty$ functions is $C^\infty$ . Mar 6, 2021 at 0:55
• Do you agree with my edits? Mar 6, 2021 at 2:20

Hint: Only noticing that all functions involved are well defined and $$C^\infty$$, should be enough. For the jacobian you need to find the partial derivatives and form the following matrix evaluated at the given point $$J(0,1,1)=\left[\begin{array}{ccc} \frac{\partial u}{\partial x} & \frac{\partial u}{\partial y} & \frac{\partial u}{\partial z}\\ \frac{\partial v}{\partial x} & \frac{\partial v}{\partial y} & \frac{\partial v}{\partial z} \end{array} \right]_{(0,1,1)}$$ where $$u=\sin(x+z)+\log(yz^2)$$ and $$v=e^{x+z} +yz$$.