I have the function $ f:]0, \pi[\times ]0,2\pi[\to \mathbb{R}^3 $ given by $$ f(x,y):=\begin{pmatrix}(R+r\cos(y))\cos(x)\\(R+r\cos(y))\sin(x)\\r\sin(y)\end{pmatrix} $$ and $ R>r>0 $ are fixed values. I want to examine if $ f $ is an immersion. If $ f $ is an immersion then $ f $ is continuos and differentiable and the derivative of $ f $ is injective.
At first I calcualted the jacobian matrix of $ f $ for all $ (x,y)\in ]0,\pi[\times ]0,2\pi[ $:
$$ Df(x,y)=\begin{pmatrix}-(R+r\cos(y))\sin(x)&-r\sin(y)\cos(x)\\(R+r\cos(y))\cos(x)& -r\sin(y)\sin(x)\\0&r\cos(y)\end{pmatrix}. $$
I defined a new function with compoments of the derivative by $ g:]0, \pi[\times ]0,2 \pi[\to \mathbb{R}^5 $ with
$$g(x,y):=\begin{pmatrix}-(R+r\cos(y))\sin(x)\\(R+r\cos(y))\cos(x)\\-r\sin(y)\cos(x)\\-r\sin(y)\sin(x)\\r\cos(y)\end{pmatrix}.$$ I want to examine if $ f $ is injective. So I define a new function $ F:]0, \pi[^2\times ]0,2 \pi[^2 $ with
$$F(x_1,x_2,y_1,y_2):=g(x_1,y_1)-g(x_2,y_2)=\begin{pmatrix}-(R+r\cos(y_1))\sin(x_1)+(R+r\cos(y_2))\sin(x_2)\\(R+r\cos(y_1))\cos(x_1)-(R+r\cos(y_2))\cos(x_2)\\-r\sin(y_1)\cos(x_1)+r\sin(y_2)\cos(x_2)\\-r\sin(y_1)\sin(x_1)+r\sin(y_2)\sin(x_2)\\r\cos(y_1)-r\cos(y_2)\end{pmatrix}$$ So I have to consider $ F(x_1,x_2,y_1,y_2)=(0,0,0,0,0)^T $. But to be honest I have no idea where I should start to solve this huge system of equations.