Suppose we have a function $f : \Bbb R^3\to\Bbb R^2$ such that $f:(x,y,z)\to(x+y^2,xy^2z) $
What's the differential of $f$?
The result of my attempt : $D_f(x,y,z)(h,k,l)=(h+2yk,y^2zh+2xyzk+xy^2l)$
What I did was simply calculating the partial derivative of each function ( $f_1=x+y^2$ and $f_2=xy^2z$ ) but I'm not sure if that's the right way to go ( as in writing the result in a couple like I did). Is the reasoning correct?
Yes, that is correct. In general the default format is column vectors, so that $$ f([x,y,z]^T)=\pmatrix{x+y^2\\xy^2z} $$ and $$ J_f([x,y,z]^T)\pmatrix{h\\k\\l}=\pmatrix{h+2yk\\y^2zh+2xyzk+xy^2l} =\pmatrix{1&2y&0\\y^2z&2xyz&xy^2}\pmatrix{h\\k\\l} $$