TITLE: $f\colon \Bbb R^3 \to \Bbb R^3 $ be defined by $f(x_1,x_2,x_3)=....$
QUESTION [5 upvotes]: I am stuck on the following problem:  

Let $f\colon \Bbb R^3 \to \Bbb R^3 $ be defined by $f(x_1,x_2,x_3)=(x_2+x_3,x_3+x_1,x_1+x_2).$ Then the first derivative of $f$ is :
  1.not invertible anywhere
  2.invertible only at the origin
  3.invertible everywhere except at the origin
  4.invertible everywhere.   

My problem is I do not know how to calculate the derivative of $f$. Can someone point me in the right direction?

REPLY [4 votes]: HINT: the derivative is a linear transformation $f':\mathbb{R^3}\rightarrow\mathbb{R}^3$ such that its matrix is
$$[f']=\left( \begin{array}{ccc}
\frac{\partial f_1}{\partial x_1} & \frac{\partial f_1}{\partial x_2} & \frac{\partial f_1}{\partial x_3} \\
\frac{\partial f_2}{\partial x_1} & \frac{\partial f_2}{\partial x_2} & \frac{\partial f_2}{\partial x_3} \\
\frac{\partial f_3}{\partial x_1} & \frac{\partial f_3}{\partial x_2} & \frac{\partial f_3}{\partial x_3}
\end{array} \right)$$

REPLY [2 votes]: The question is most likely looking for the Jacobian:
$$J=f'(x_1,x_2,x_3)=\begin{bmatrix}\frac{\partial f_1}{\partial{x_1}} & \frac{\partial f_1}{\partial{x_2}} & \frac{\partial f_1}{\partial{x_3}}\\\frac{\partial f_2}{\partial{x_1}} & \frac{\partial f_2}{\partial{x_2}} &\frac{\partial f_2}{\partial{x_1}}\\\frac{\partial f_3}{\partial{x_1}} & \frac{\partial f_3}{\partial{x_2}} &\frac{\partial f_3}{\partial{x_3}}\end{bmatrix}$$
This seems like a lot of work, but since this is a linear transformation, the Jacobian will be identical to the transformation matrix itself.