WWJD (What would Jacobi do?)

Calculus Level 5

Suppose \(f: {\mathbb{R}}^{3} \rightarrow {\mathbb{R}}^{3} \) such that

\(f(x,y,z) = \begin{bmatrix} x + y + z \\ xy + yz + zx \\ xyz\end{bmatrix}\)

Determine the determinant of the total derivative of \(f\) at each of the points \((0,0,1),(0,1,0),(0,1,1),(1,0,0),(1,0,1),(1,1,0),\) and \((1,1,1)\).

What is the value of the sum of the determinants?

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