Skip to main content

Exercises 10.5 Homework

1.

(Theoretical problem.) Let

\begin{equation*} \mathbf F(x,y,z) =\begin{pmatrix} xyz \\ 2(xy+yz+xz) \\ x^2+y^2+z^2 \end{pmatrix} \end{equation*}

Find the Jacobian matrix of \(\mathbf F\text{.}\) (The matrix entries will involve the variables \(x,y,z\)).

2.

At which of the following points \((x,y,z)\) is the Jacobian matrix from the previous exercise invertible?

\begin{equation*} \begin{pmatrix} 1\\1\\1\end{pmatrix}, \begin{pmatrix} 3\\1\\5\end{pmatrix}, \begin{pmatrix} 2\\1\\2\end{pmatrix} \end{equation*}

To save time, use Matlab: define an anonymous function like J = @(x,y,z) [... ; ... ; ...] so that you can then do rank(J(1,1,1)) and so on, without computing the matrices yourself. Note the difference between this approach to J and the approach of Section 10.2: here the function J takes three scalar arguments, instead of one vector argument.

3.

Let \(V = 36 + 2a\text{,}\) \(A = 72 + 3a\text{,}\) and \(D = 7 + a/5\) where \(a\) is the first digit of your SUID. Write a Matlab script which uses Newton's method to find the dimensions of a rectangular box with volume \(V\text{,}\) surface area \(A\text{,}\) and space diagonal \(D\text{.}\) The computations in Problem 1 will be helpful, because a box with dimensions \(x, y, z\) has volume \(xyz\text{,}\) surface area \(2(xy+yz+xz)\text{,}\) and space diagonal \(\sqrt{x^2+y^2+z^2}\text{.}\)

Reminder: the vector function \(\mathbf F\) that you define should be such that \(\mathbf F = 0\) expresses the system of equations that you wish to solve. Since \(\mathbf F\) should be differentiable, try to set it up so that there is no square root in \(\mathbf F\text{.}\)