\\\\(
\nonumber
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Exercise 1: Repetition with Orthogonal Substitution
A symmetric matrix is given:
$$
\mathbf A= \begin{matr}{cc}
\displaystyle{\frac{288}{25}}&\displaystyle{\frac{84}{25}}\\\\\\\\
\displaystyle{\frac{84}{25}}&\displaystyle{\frac{337}{25}}
\end{matr}\,.$$
A
Find in $\,\mathbb R^{2\times 2}\,$ a positive orthogonal matrix $\,\mathbf Q\,$ and a diagonal matrix $\,\Lambda\,$ such that
$$
\Lambda=\mathbf Q^{-1} \mathbf A\, \mathbf Q\,.$$
An ellipse $\,\mathcal E\,$ is in the standard orthogonal $\,(x, y)$-coordinate system in the plane given by the matrix equation
$$
\begin{matr}{cc} x&y \end{matr} \,\mathbf A\,
\begin{matr}{c} x \\\\ y \end{matr}=144\,.$$
B
Determine the semi-axes for $\,\mathcal E\,.$
Exercise 2: Repetition with Function of Two Variables
For a smooth function $\,f:\mathbb R^2 \rightarrow \mathbb R\,$ with $\,f(0,0)=0\,$ a vector field $\,\mathbf V\,$ in the $\,(x,y)$-plane is given by
$$\mathbf V(x,y)=\nabla f(x,y)=(x-y^2+1,-2\,x\,y)\,.$$
A
Find all stationary points for $\,f\,.$
B
Determine the Hessian matrix of $\,f\,,$ and explain that $\,f\,$ has exactly one proper local minimum and no proper local maxima.
C
Determine the tangential curve integral of $\,\mathbf V\,$ along a curve $\,\mathcal K\,$ of your own choice from the origin to an arbitrary point $\,(x,y).$ Hint: You may use the formula
$$
(x,y)\cdot \int_0^1 \mathbf V(ux,uy)\,\mathrm{d}u\,.
$$
Or you can integrate along the broken line in the $\,(x,y)$-plane that connects $\,( 0,0)\,$ and $\,( x,0)\,$ and $\,( x,0)\,$ and $\,( x,y)\,.$
D
Determine the value of $\,f\,$ at the proper local minimum point you found in Question B).
Exercise 3: Repetition with Gauss and Stokes
In the $\,(x,y)$-plane in the $\,(x,y,z)$-space, a set
$$\,A=\left\{\,(x,y)\,|\,0\leq x \leq2\,\,\mathrm{and} -\frac{\pi}{2}\leq y \leq\frac{\pi}{2}\,\right\}\,$$
and a function
$$h(x,y)=x\cos(y)\,$$
are given. Let $\,\mathcal F\,$ denote the part of the graph of $\,h\,$ that is vertically above $\,A,$ see the figure.
A
Determine a parametrization $\,\mathbf r(u,v)\,$ for $\,\mathcal F,$ and determine the nomal vector
$$\mathbf N(u,v)=\mathbf r'_u(u,v) \times \mathbf r'_v(u,v)\,$$
corresponding to $\,\mathbf r(u,v)\,$.
Suppose $\,\mathbf V\,$ is a vector field int the $\,(x,y,z)$-space and that
$$\,\mathrm{Div}(\mathbf V)(x,y,z)=x+y+z\,\,\,\,\mathrm{and}\,\,\,\,\mathbf{Curl}(\mathbf V)(x,y,z)=(3z,3x,3y)\,.$$
B
Find the tangential curve integral (circulation) of $\,\mathbf V\,$ along the closed boundary curve $\,\partial \mathcal F\,$ of $\,\mathcal F\,$ where the orientation of $\,\partial \mathcal F\,$ is as shown in the figure above.
C
Let $\, \Omega\,$ denote the 3-dimensional solid region that lies vertically between $\,A\,$ and $\,\mathcal F\,.$ Determine the flux of $\,\mathbf V\,$ out of the closed surface $\,\partial \Omega\,$ of $\, \Omega\,.$
