Mathematical Methods of Physics 1

Differential and integral calculus for functions with one variable: Differentiation rules, Taylor expansion, Bernoulli-L’Hospital rule, curve sketching, numerical differentiation, integration rules, numerical integration, elliptical integration.

Differential and integral calculus for functions with multiple variables: Recapitulation of vector calculus, scalar fields, vector fields, partial differentiation, gradient, total differential, directional derivative, expanded chain rule, Taylor expansion, relative extreme values of functions with multiple variables, curves in Rⁿ, line integral, path independency and antiderivative, double and surface integrals, trifold and volume integrals, fundamentals of vector analysis (gradient, divergence, rotation).

After the successful participation in the module the student is able to:

1. master and apply the most important rules of differential calculus
2. know and apply the most important rules of integral calculus
3. know the possibilities of numerical integration
4. master the fundamentals of vector calculus
5. apply differentiation and integration to functions with multiple variables
6. describe the fundamentals of vector analysis.

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lecture: teacher-centred teaching

Exercise: The exercise is held in small groups. In the weekly session exercises are presented by the students and the tutor. They also provide room for discussions and additional explanations to the lectures.

writing on blackboard, presentation

Mathematische Hilfsmittel der Physik, W. Kuhn, H. Stöckel und H. Glaßl, Johann Ambrosius Barth Verlag, Heidelberg, Leipzig, 1995

Mathematische Methoden in der Physik, C. B. Lang, N. Pucker, Spektrum Akademischer Verlag, Heidelberg, Berlin, 1998

Der mathematische Werkzeugkasten – Anwendungen in der Natur und Technik, G. Glaeser, Spektrum Akademischer Verlag, Heidelberg, Berlin, 2004

There will be a written exam of 90 minutes duration. Therein the achievement of the competencies given in section learning outcome is tested exemplarily at least to the given cognition level using calculation problems and comprehension questions.

For example an assignment in the exam might be:

• Differentiation and Integration of a given function f(x).
• Taylor expansion and calculation of the integral to a given function f(x), integration using Taylor expansion.
• Calculation of gradient and total differential of a given skalar field Φ(x,y,z).
• Give a criterion for the path independence of line integrals over a given vektor field V(x,y,z).
• Calculation of center of mass for curved lines, surfaces, and 3D bodies (e.g. a line segment, a segment of the surface of a sphere, a segment of a solid sphere).

Participation in the exercise classes is strongly recommended since the exercises prepare for the problems of the exam and rehearse the specific competencies.