This volume provides the essential mathematical foundations for undergraduate studies in physics. The content corresponds to a four-semester mathematics course. The structure is oriented towards what physicists need as early as possible, such as vector calculus, differential and integral calculus, as well as elementary differential equations. By motivating the mathematical concepts and justifying most results, the aim is to convey to readers the mathematical working methods and the essential problem understanding for mathematical physics. The effectiveness of the mathematical methods presented here is demonstrated through numerous applications to physics.