Learn · Multivariable Differential Calculus and Ordinary Differential Equations
Lecture 58 of 60Week 12 Continuous or Smooth Dependence on Parameters
Continuous Dependence on Parameters
Prof. Vamsi Pritham Pingali · IISc Bangalore
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- Multivariable Differential Calculus and Ordinary Differential Equations, IISc Bangalore (at NPTEL)
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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