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In class, we defined the fundamental vector product of $r$, where $$r(u,v) = (X(u,v), Y(u,v), Z(u,v)),$$ as shown in the image. I understand how we got everything in the first line, but how are we going from the three determinants at the end of the first line to each of the coefficients in the second line? And in particular, what does $\frac{\partial (Y,Z) }{\partial (u,v)}$ represent? Formula

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$\frac{\partial(Y,Z)}{\partial(u,v)}$ represents the Jacobian of $(Y,Z)$ wrt $(u,v)$ and it is equal the determinant as in the answer. For more details please refer https://en.wikipedia.org/wiki/Jacobian_matrix_and_determinant

For example, if $x = x(u,v)$ and $y = y(u,v)$ are two functions of variables $u$ and $v$, then the Jacobian determinant of $(x,y)$ w.r.t $(u,v)$ is,

$J = \frac{\partial(x,y)}{\partial(u,v)} = \begin{vmatrix} \frac{\partial x}{\partial u} & \frac{\partial y}{\partial u}\\ \frac{\partial x}{\partial v} & \frac{\partial y}{\partial v}\end{vmatrix}$

Similarly it works for $n$ functions of $n$ variables

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