Find the Jacobi-matrix of the following functions

\(\displaystyle f(x,y)=\big(x+y,xy,\cos(x+y)\big); \quad g(x,y)=\big(e^{x+y},xy\big); \quad h=f\circ g. \)

Difficulty: 1.

Prove that vectorial product viewed as a \(\displaystyle \R^6\to\R^3\) function is differentiable. What is its derivative?

Difficulty: 2.

Find the Jacobi-matrix of the following functions:

\(\displaystyle f(x,y)=\big(\sin x, \cos y); \quad g(x,y)=\big(\log x,x^2+y^2); \quad h=f\circ g. \)

Difficulty: 2.

\(\displaystyle f:\R^2\to\R^3\), \(\displaystyle (x,y)\mapsto (e^x,x^2+y^2,\sin x)\); \(\displaystyle g:\R^3\to\R\), \(\displaystyle (X,Y,Z)\mapsto XY\). \(\displaystyle (g\circ f)'=?\)

Difficulty: 3.

What is the Jacobi matrix of the local inverse of \(\displaystyle f(x,y) = (x^2-y^2,2xy)\)?

Difficulty: 4.

Let \(\displaystyle f: \R^p \rightarrow\R^q\) be differentiable at the points of the interval \(\displaystyle [a,b]\subset\R^p\). Prove that

\(\displaystyle |f(b)-f(a)| \le |b-a| \cdot \sup_{c\in[a,b]} ||f'(c)||. \)

Difficulty: 4.

Let \(\displaystyle A:\R^n\to \R^n\) be an invertible linear transformation. Show that

\(\displaystyle ||A^{-1}||={1\over \min \{Ax|x\in S^{n-1}_0(1)\}}.\)

Difficulty: 5.

Find an \(\displaystyle A:\R^n\to\R^n\) linear transformation for which

\(\displaystyle \sqrt{\sum_{i,j}a_{i,j}^2} > ||A||.\)

Show that \(\displaystyle \geq\) is always true.

Difficulty: 5.

(a) Prove that all linear maps \(\displaystyle \R^p\to\R^q\) are Lipschitz.

(b) Prove that if \(\displaystyle A\in\mathrm{Hom}(\R^p,\R^p)\) is invertible, then \(\displaystyle \exists c>0 \, \forall x\in\R^p \, |A(x)| \ge c|x|\).

Difficulty: 5.

Prove that for all \(\displaystyle A\in\mathrm{Hom}(\R^p,\R^p)\) \(\displaystyle ||A||\ge|\det A|^{1/p}\).

Difficulty: 7.

Prove that

\(\displaystyle \max_{1\le j\le p} \sqrt{\sum_{i=1}^q a_{ij}^2} \le \left\Vert\begin{pmatrix} a_{11}&\ldots&a_{1p} \\ \vdots&&\vdots\\ a_{q1}&\ldots&a_{qp} \\ \end{pmatrix}\right\Vert \le\sqrt{\sum_{i=1}^q\sum_{j=1}^pa_{ij}^2}. \)

Give an example when equality does not hold.

Difficulty: 8.