This resource was developed by [Nils Tilton][1], Leah Berman, and Corey R. Randall at the Colorado School of Mines partly through grant funding from the Colorado Department of Education and Arthur Lakes Library. Abstract: At the graduate level, engineering students face several challenges when it comes to mathematical proficiency. First, math and physics in graduate courses and research become inextricably tied, and require more intuition and creativity than at the undergraduate level. The problems are more open-ended, and not easily solved using the “follow-the-recipe” approach that rote-learning tends to promote. Some universities address this issue by offering courses dedicated to mathematical modeling. A google search for “mathematical modeling” returns a host of lecture notes and textbooks that we encourage students to consult. The second challenge is that graduate courses and research require advanced math that draws liberally from differential, integral, and vector calculus, as well as linear algebra, complex numbers, and integral transforms such as Fourier and Laplace transforms. The third challenge is that real-world systems studied at the graduate level often produce mathematical models that cannot be solved analytically. This requires students to become proficient in numerical methods and computer programming. This text has three primary objectives. The first is to provide students an introduction to the advanced mathematical concepts they are likely to encounter in their courses and research. We focus on the topics of ordinary differential equations, partial differential equations, and Fourier transforms because these are used broadly in solid mechanics, material science, thermal-fluid sciences, dynamics, automation, and control. As we present these topics, we also make an effort to review basic concepts from calculus and linear algebra. Our second objective is to simultaneously give students a rudimentary experience in basic numerical methods and computer coding with MATLAB. We focus on numerical integration and differentiation and solving ordinary and partial differential equations using finite di↵erence methods. Finally, we make an e↵ort to develop homework assignments, projects, and exams with open-ended problems requiring mathematical modeling. While an undergraduate math homework or exam typically asks students to solve a given equation, this courses presents students with physical problems for which they must first build the mathematical model before they can solve it. [1]: https://mechanical.mines.edu/project/tilton-nils/