Let \(\vec{p} = (2, 4\pi, 1) \in \mathbb{R}^3\) and let \(f: \mathbb{R}^3 \to \mathbb{R}\) be a \(C^1\) function such that
\(f(\vec{p}) = 3\), \(\frac{\partial f}{\partial x_1}(\vec{p}) = 5\), \(\frac{\partial f}{\partial x_2}(\vec{p}) = 5\), \(\frac{\partial f}{\partial x_3}(\vec{p}) = 6\)
Calculate the trace of the Jacobian matrix \(tr(DF)\) evaluated at the point \(\vec{p}\) for the function \(F: \mathbb{R}^3 \to \mathbb{R}^3\) defined by
\(F(\vec{x}) = \begin{bmatrix} x_1^2 + x_2^3 \\ -x_1^3 x_3 \cos(3x_2) \\ f(\vec{x}) \end{bmatrix}\)
(Recall that the trace of a matrix is the sum of its diagonal entries.)
