Advanced Calculus

This course provides a comprehensive introduction to multivariable calculus and ordinary differential equations through the lens of mathematical modeling and computation. The course begins with the mathematical representation of curves and surfaces using Cartesian, cylindrical, spherical, and parametric forms, followed by the study of smoothness, continuity, differentiability, gradients, directional derivatives, and geometric properties of surfaces. It then develops techniques for measuring geometric and physical quantities, including curve length, surface area, volume, mass, and averages using single, double, and triple integrals.

The course concludes with analytical and numerical methods for solving ordinary differential equations and systems of differential equations arising from real-world applications. Throughout the course, Python-based computational tools are integrated to visualize mathematical objects, perform symbolic and numerical computations, and explore applications in science and engineering.

The course opens with the language of multivariable functions, spaces, and analytical geometry, then moves into three-dimensional shapes and coordinate systems. From there, students study limits, derivatives, and gradients, and put these tools to work in optimization problems, including constrained optimization with Lagrange multipliers.

The middle of the semester is given to multidimensional integrals and their applications: computing lengths, areas, volumes, and other quantities drawn from real-world modeling. Vector-valued functions and vector fields follow, along with the major theorems of vector calculus. The course closes with sequences, series, and an introduction to differential equations, rounding out a complete first foundation in the mathematics that underlies most quantitative fields. Python-based computation runs throughout the semester, turning each abstract result into something students can plot, test, and adjust.

By the end of this course, students will have had the opportunity to:

• Model and plot curves and surfaces in Cartesian, polar, cylindrical, and spherical coordinate systems.
• Analyze and visualize continuity, partial derivatives, and differentiability of multivariable functions.
• Compute directional derivatives and gradients, and apply them to optimization problems, including extrema and Lagrange multipliers.
• Compute lengths of curves, areas of surfaces, and volumes of regions using single, double, and triple integrals.
• Analyze the motion of a particle in space using vector-valued functions, including velocity, acceleration, curvature, and torsion.
• Determine whether a vector field is conservative, and compute line integrals, circulation, and flux using Green's, Stokes', and the Divergence Theorems.
• Approximate functions using Taylor series expansions.

By the end of this course, each student will have had the opportunity to:

• Model curves and surfaces in Cartesian, polar, cylindrical, and spherical coordinates, and plot them graphically | Apply Outcome
• Analyze limits, continuity, and differentiability of multivariable functions, and compute partial derivatives | Analyze Outcome
• Apply gradients, directional derivatives, and Lagrange multipliers to solve optimization problems | Apply Outcome
• Calculate lengths, areas, and volumes using single, double, and triple integrals | Apply Outcome
• Describe motion in space using vector-valued functions, including velocity, acceleration, curvature, and torsion | Comprehend Outcome
• Evaluate conservative vector fields, and compute line integrals, flux, and circulation using key theorems: Green's, Stokes', and Divergence | Analyze Outcome
• Approximate functions using Taylor series expansions for applications | Apply Outcome

• J. Hass, C. Heil, and M. D. Weir, Thomas' Calculus, 15th Edition, Pearson Education, 2024.

• E. Kreyszig, Advanced Engineering Mathematics, 10th Edition, John Wiley and Sons, 2011.
• T. M. Apostol, Calculus, Volumes I and II, 2nd Edition, John Wiley and Sons, 1967 and 1969.

All Freshmore courses are graded on a relative basis. For Advanced Calculus, this currently breaks down as:

• Mid-Semester Exam (90 minutes): 20%
• Two Assignments with Viva Voce: 20%
• Lab Evaluation: 20%
• Final Comprehensive Exam (180 minutes): 40%, made up of 20% closed book and 20% open book

The nature and weight of these components are tentative and may be revised.

Students who miss an evaluation component for a genuine reason, such as a medical emergency, a personal or family emergency, or an official university engagement, may apply for a makeup. Applications go to the instructor of record along with supporting documentation. The instructor of record decides whether to award the makeup, and it is confirmed once the Student Life team or the Office of Academic Affairs certifies the case.