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MATH 103: Multivariate Calculus for Machine Learning
Course Code MATH 103
Course Name Multivariate Calculus for Machine Learning
Department Mathematics
Semester Offered Odd (Term 1)
Tuition Hours 21 hours
Course Level Foundational
Pre-requisite MATH 102: Calculus for Machine Learning
Co-requisite MATH 101: Linear Algebra for Machine Learning
Course Objective Single-variable calculus tells you how one thing changes. That is not enough. Real systems do not depend on one variable. They depend on hundreds or thousands of parameters interacting together.

This course extends calculus into that reality. Students will learn how change works in high-dimensional systems, how gradients guide optimization across many parameters, and how modern machine learning models actually navigate complex loss landscapes.

By the end, students should be able to look at a training process and understand how multiple parameters are being updated together, not just mechanically but conceptually. This is where calculus stops being academic and starts becoming the engine behind real learning systems.
Course Philosophy This course emphasizes
  • Thinking in many dimensions, not just computing formulas
  • Understanding gradients as directions, not just symbols
  • Optimization as a system-level process
We avoid unnecessary formalism. Every concept must connect to how real models are trained and tuned. If students cannot relate it to their AI product, we have gone too far.
Course Learning Outcomes Upon successful completion of this course, students will be able to:
  • Work with functions of multiple variables and visualize their behavior.
  • Compute partial derivatives and interpret their meaning in real systems.
  • Understand gradients as directions of steepest ascent or descent in high-dimensional spaces.
  • Apply gradient-based optimization to multi-parameter systems.
  • Understand Jacobians at a conceptual level, especially in transformations between spaces.
  • Reason about constrained optimization problems, common in real-world ML systems.
  • Debug and tune models more effectively by understanding how parameters interact.
Course Author Sagar Udasi
MSc Statistics and Data Science with Computational Finance from The University of Edinburgh.
Contact: sagar.l.udasi@gmail.com
Course Organiser TBD
No. Lecture Title Concepts Covered Lecture Objective
01 When One Variable Is Not Enough Multivariable functions, surfaces Students understand why real ML models depend on many variables, not just one.
02 Visualizing Loss in 3D and Beyond Surface plots, intuition of high-dimensional spaces Helps students build intuition for optimization landscapes used in training.
03 Partial Derivatives: Changing One Thing at a Time Partial derivatives, interpretation Students learn how individual parameters affect output in a system.
04 Gradients: The Direction That Actually Matters Gradient vector, geometric meaning Core concept for optimization in ML. Students understand direction of steepest change.
05 Gradient Descent in High Dimensions Multi-variable optimization Extends earlier knowledge to real ML systems with many parameters.
06 When Optimization Gets Tricky Saddle points, flat regions Prepares students to deal with real-world training difficulties.
07 The Chain Rule Scales Up Multivariable chain rule Foundation for understanding backpropagation in deep networks.
08 What Backpropagation Really Does in High Dimensions Gradient flow through layers Connects multivariate calculus directly to neural network training.
09 Jacobians Without the Fear Jacobian matrix, transformations Helps students understand how outputs change with respect to multiple inputs.
10 Constraints Change Everything Constrained optimization, Lagrange multipliers (intuition) Introduces real-world limitations in optimization problems.
11 Regularization: Controlling the Model Penalties, overfitting intuition Connects calculus to better generalization in ML systems.
12 From Math to Code Implementing multivariate gradient descent Students translate theory into working optimization routines.
13 Lab: Training a Multi-Parameter Model Hands-on optimization Students apply concepts to a real training loop.
14 Project Integration: Tuning Your AI System Applying optimization to product Students directly improve their Term 1 AI agent using learned concepts.
Component Weightage
Written Examination (2 hours) 50%
Practical Assignment (Optimization Task) 30%
Project Integration (Applied to AI Product) 20%
Type Resource Provider
Lecture Multivariable Calculus MIT OpenCourseWare
Lecture Essence of Calculus (Multivariable Sections) 3Blue1Brown
Reading Calculus: Early Transcendentals (Multivariable Chapters) James Stewart
Practical CS231n Optimization and Backprop Notes Stanford University