Learn · Multivariable Differential Calculus and Ordinary Differential Equations
Lecture 10 of 60Week 2 Review of Some Prerequisites (Topology of Euclidean Spaces and Limits and Continuity of Scalar and Vector Fields)
Detailed Review of Algebra of Limits and Algebra of Continuous Functions
Prof. Vamsi Pritham Pingali · IISc Bangalore
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- From
- Multivariable Differential Calculus and Ordinary Differential Equations, IISc Bangalore (at NPTEL)
- Licence
- CC BY-NC-SA, as NPTEL states it. Non-commercial use only.
- Attribution
- NPTEL states: "Distributed under Creative Commons Attribution-ShareAlike / CC BY - NC - SA", beside a BY-NC-SA badge.
- Who can watch
- Tweens profiles and older. How courses are placed
All the lectures
- 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
- 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
- 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
- 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
- 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
- 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.)
- 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.)
- About the Constraining Sets in the Lagrange Multiplier Method
- Introduction to Manifolds Embedded in R^n
- Gradient Descent
- Gradient Descent (Contd.)
- Gradient Descent (Contd.)
- 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
- Triangularization Theorem
- Exponential of Square Matrices
- Properties of Matrix Exponential
- Properties of Matrix Exponential (Contd.)
- Solving Homogeneous and Inhomogeneous Linear Systems
- 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
- Improvement of Picard's Existence Theorem
- Characterization of Maximal Interval of Existence
- Continuous Dependence on Parameters
- Smooth Dependence on Parameters
- Smooth Dependence on Parameters (Contd.), and Numerical Methods to Solve ODEs