# CAM oral exam

## Informations about the oral CAM exam

• The CAM admission commission schedules an oral exam, if it is unclear if the knowledge of some candidate is sufficient. For candidates with non-mathematical degrees an oral exam is prescribed without exception.
• The oral exam has to check that the knowledge of a candidate is sufficient for a successful CAM study.
• Basic knowledge (absolutely necessary) [see Footnote *]:
• Good knowledge in Analysis: norms, sequences, series, differentiation and integration, both also in a multi-dimensional setting, ordinary differential equations. We want to see especially also a good understanding of theoretical results like the Picard-Lindelöf theorem of existence and uniqueness of ordinary differential equations (ODEs).
• Good knowledge of linear algebra: vector spaces, linear mappings, determinants, eigenvalue theory.
• Additional knowledge [see Footnote *]:
1. Basic knowledge of functional analysis: Banach spaces, Hilbert spaces, examples like Lp- or Sobolev spaces
2. Basic knowledge of partial differential equations: examples of PDEs, classification (elliptic, parabolic, hyperbolic), existence and uniqueness of solutions.
3. Basic knowledge of numerical analysis: Solution of linear and nonlinear systems, interpolation, quadrature, numerical solution of ODEs.
4. Basic knowledge in optimization: optimization problems, solution methods
• The CAM oral exam should cover topics in one of the additional knowledge domains 1 and 2 and one in the domains 3 and 4. It can also extend to other subjects like the candidate’s Bachelor thesis.
• Here are some sample questions: [see Footnote **]
• What is a Banach space? Give an example.
• What is a PDE? What is the difference between ODE and PDE? Give an example for a PDE. What is its type? Under which conditions do we have solutions? When are those uniquely determined?
• What is the general form of an ordinary differential equation? Under which conditions do we have solutions? When are those uniquely determined? How could they be approximated numerically? Solve the simple ODE u'(x)=…
• How would you solve the nonlinear system of equations F(x)=0 numerically? How does this look like for the special choice F(x)=…?
• Let f(x)=expression-in-x. How can I find an extremum (minimum/maximum) of f? What if we have a side condition g(x)=… which has to be fulfilled?

Footnotes:

• [*] Note that this is a purely personal list of topics which is not complete as well. It is intended to give you an impression what I (Nicolas Neuss) personally consider relevant. Other examiners may vary.
• [**] Note that these are sample questions which I (and even more other examiners) may or may not pose.