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The history is that in Frankfurt Walter Greiner started with theoretical physics in the 1st semester, which was a novum in these days. That's why there had to be "Mathematical Methods" in the 1st-semester theory lecture. This is something you have to offer anyway, because the math lectures, which IMHO should be taken by the physicists together with the mathematicians, cannot deliver the necessary mathematical tools as quickly as needed in the theoretical-physics lectures.
Math must be taught in a rigorous manner with theorems and strict proofs. "Mathematical Methods" just provides plausibility arguments for the calculational machinery needed for theoretical physics. Thus the lecture mainly concentrates on vector algebra and vector calculus (assuming that the students are familiar with differentiation and integration for functions with one real variable, which is not always fulfilled either nowadays though, because the German high-school teaching is in a monotonic decline particularly in math) and on the calculational side of the subject, i.e., usually no rigorous proofs for theorems (e.g., Stokes's and Gauss's integral theorems) are given but plausibility arguments.
Usually that's done with examples from physics, i.e., classical mechanics and some electrostatics. This also has the advantage that early on you learn to think in the typical physicists' way in terms of mathematics as a language to formulate and work with the physical laws.
Math must be taught in a rigorous manner with theorems and strict proofs. "Mathematical Methods" just provides plausibility arguments for the calculational machinery needed for theoretical physics. Thus the lecture mainly concentrates on vector algebra and vector calculus (assuming that the students are familiar with differentiation and integration for functions with one real variable, which is not always fulfilled either nowadays though, because the German high-school teaching is in a monotonic decline particularly in math) and on the calculational side of the subject, i.e., usually no rigorous proofs for theorems (e.g., Stokes's and Gauss's integral theorems) are given but plausibility arguments.
Usually that's done with examples from physics, i.e., classical mechanics and some electrostatics. This also has the advantage that early on you learn to think in the typical physicists' way in terms of mathematics as a language to formulate and work with the physical laws.