Course11110800260 lectures
Multivariable Differential Calculus and Ordinary Differential Equations
Prof. Vamsi Pritham Pingali · IISc Bangalore
About
This course starts with a reasonably detailed review of linear algebra and topology in Rn. Then we shall cover multivariable differential calculus. We shall also introduce manifolds via the implicit function theorem. We shall then cover an application or two to Optimisation/AI/ML – namely gradient descent (the continuous and discrete versions) and the backpropagation algorithm. The next few weeks will deal with ordinary differential equations, in particular solving linear systems, and existence (and dependence on parameters) theory for nonlinear ones. The latter part involves concepts of multivariable calculus. The course is meant for students of pure and applied mathematics, AI, CS, physics, and engineering.
Intended audience: Undergraduates and Graduate students of mathematics, computer science, physics, and engineering
Prerequisites: One-variable real analysis (at the level of Rudin or Terence Tao, including metric spaces), Linear algebra (up to and including diagonalisation of Hermitian matrices).
Lectures
Week 1 Review of Some Prerequisites (Linear Algebra and Topology of Euclidean Spaces)
- Vector Spaces and Subspaces, Linear Span and Linear Independence, Linear Transformations
- The Matrix Representation of a Linear Transformation
- Systems of Linear Equations and Gauss-Jordan Elimination Method
- The Row and Column Spaces of a Matrix, Rank and Determinant
- Inner Product Spaces and Normed Vector Spaces
Week 2 Review of Some Prerequisites (Topology of Euclidean Spaces and Limits and Continuity of Scalar and Vector Fields)
- Eigenvalues, Eigenvectors and Diagonalizability
- Interior Points and Open Sets, Convergent Sequences and Continuous Functions
- Closed Sets and Limit Points, Connectedness and Compactness, Limits and Continuity
- Further Discussion on Limits and Continuity
- Detailed Review of Algebra of Limits and Algebra of Continuous Functions
Week 3 Introduction to Multivariable Differentiability
- Directional Derivatives and Partial Derivatives
- Differentiability of a Multivariable Function
- Differentiability of Vector Fields, Gradient and Jacobian Matrix
- Continuous Differentiability
- Algebra of Differentiable Functions and Some Other Properties of Differentiability
Week 4 Chain Rule, LMVT, and Mixed Partial Derivatives
- More Discussion On Differentiation Of Several Variable Functions, and Derivative Computation For Some Functions
- Chain Rule
- Further Discussion On Chain Rule
- LMVT in Multivariable Calculus, and Directional Derivative Along Regular Paths
- Further Discussion On Directional Derivative, and Mixed Partial Derivatives
Week 5 Clairaut's Theorem, Taylor's Theorem, and Second Derivative Test For Multivariable Functions
- Clairaut's Theorem
- Taylor's Theorem and Second Derivative Test in Single Variable Functions
- Taylor's Theorem For Multivariable Functions
- Second Derivative Test For Multivariable Functions
- Further Discussion on Second Derivative Test
Week 6 Inverse Function Theorem (IFT) For Several Variables
- Discussion about convergence of Taylor series and bump functions
- Inverse Function Theorem for one variable functions
- Proof of the Inverse Function Theorem for several variables
- Proof of the Inverse Function Theorem for several variables (Contd.)
- Proof of the Inverse Function Theorem for several variables (Contd.)
Week 7 Implicit Function Theorem
- The Implicit Function Theorem for Functions of Two Variables
- The Implicit Function Theorem for Functions of Several Variables
- Some Further Applications of the Implicit Function Theorem
- Constrained Optimization Problem and the Lagrange Multiplier Method
- The Lagrange Multiplier Method (Contd.)
Week 8 Embedded Manifolds, Gradient Descent Algorithm
- About the Constraining Sets in the Lagrange Multiplier Method
- Introduction to Manifolds Embedded in R^n
- Gradient Descent
- Gradient Descent (Contd.)
- Gradient Descent (Contd.)
Week 9 Applications of Gradient Descent (Backpropagation), Introduction to ODEs
- Some Applications of Gradient Descent
- Backpropagation (contd.), Introduction to ODEs
- Some Applications of ODEs to Real-World Problems
- Systems of ODEs (contd.)
- Methods of Solving a First-Order System of ODEs
Week 10 Exponential of Square Matrices and Solving Linear Systems
- Triangularization Theorem
- Exponential of Square Matrices
- Properties of Matrix Exponential
- Properties of Matrix Exponential (Contd.)
- Solving Homogeneous and Inhomogeneous Linear Systems
Week 11 Existence and Uniqueness Theory for ODEs
- Uniqueness of Solutions to Inhomogeneous Linear System
- Existence of Solutions to Inhomogeneous Linear System
- Uniqueness of Solutions to ODEs in General Form
- General Existence Theory for ODEs
- Picard's Existence Theorem for ODEs
Week 12 Continuous or Smooth Dependence on Parameters
Syllabus and materials
At NPTEL, as the course publishes them.
Credits and licence
- Institution
- IISc Bangalore
- Department
- Mathematics
- Teacher
- Prof. Vamsi Pritham Pingali
- Level
- Graduate
- Language
- English
- Subjects
- Mathematics
- Source
- NPTEL
- Licence
- CC BY-NC-SA, as NPTEL states it. Non-commercial use only. NPTEL's terms
- Attribution
- NPTEL states: "Distributed under Creative Commons Attribution-ShareAlike / CC BY - NC - SA", beside a BY-NC-SA badge.
- Who can watch
- NPTEL gives no rating. Invisible Network shows it to Tweens profiles and older, judged from what the source says about it. How courses are placed
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