This book addresses the most important mathematical topics from lectures on theoretical physics. It complements the mathematical foundations that physics students receive in their courses on analysis and linear algebra. While the author often omits detailed proofs, he does not compromise on the precise formulation of the concepts. Numerous physical examples aim to promote a thorough understanding of the mathematical relationships and to reveal their significance for our understanding of nature. A collection of exercises at the end of each chapter provides readers with the opportunity to practice. This section covers ordinary differential equations, Fourier analysis, and coordinate-free differential calculus in vector spaces, thereby mathematically supporting the theoretical lectures on mechanics.