Math IV: Multivariate Calculus¶

• Chain Rule on a Curve
• Multivariable Chain Rule
• Second-Order Partial Derivative
• Steepest Ascent Direction
• Twice Continuous Differentiability on $\mathbb{R}^n$
• Velocity Vector of a Curve
• Hessian Matrix
• Symmetry of Second Derivatives

Review week 1 in Concept Finder 8 concepts

• Exogenous and Endogenous Variables
• Higher-Order Partial Derivative
• Higher-Order Symmetry of Partials
• Implicit Function Theorem
• Multivariate Taylor Approximations
• Partial Jacobians
• Third-Order Derivative Tensor
• Tensor Contraction for Taylor Terms

Review week 2 in Concept Finder 8 concepts

• Independent Linear Constraint System
• Left and Right Inverses
• Local Injectivity, Surjectivity, and Bijectivity
• Multivariable Inverse Function Theorem
• Principal Minors and Sylvester's Criterion
• Quadratic Form on a Constraint Set
• Restriction of a Function
• Bordered Matrix Test
• Jacobian Rank Local Behaviour Theorem

Review week 3 in Concept Finder 9 concepts

• Convex Set in $\mathbb{R}^n$
• Global Maximum and Minimum on $\mathbb{R}^n$
• Lagrangian Function
• Local Maximum and Minimum on $\mathbb{R}^n$
• Nondegenerate Constraint Qualification
• Saddle Point
• Unconstrained Extrema Conditions
• Convex and Concave Functions on $\mathbb{R}^n$
• Hessian Test for Convexity and Concavity
• Lagrange Multiplier
• Method of Lagrange for Equality Constraints

Review week 4 in Concept Finder 11 concepts

• Binding and Nonbinding Constraints
• Complementary Slackness
• Lagrangian Function
• Nondegenerate Constraint Qualification
• Method of Lagrange for Inequality-Constrained Maximization
• Method of Lagrange for Mixed Constraints
• Method of Lagrange for Constrained Minimization

Review week 5 in Concept Finder 7 concepts

• Bordered Hessian Matrix
• Envelope Theorem for Constrained Optimization
• Lagrange Multiplier
• Parametric Constrained Optimization Problem
• Second-Order Sufficient Conditions for Constrained Optimization
• Value Function
• Shadow Price in Constrained Optimization

Review week 6 in Concept Finder 7 concepts

• General Method of Lagrange
• Kuhn-Tucker Conditions with Nonnegativity Constraints
• Lagrangian Function
• Nonnegativity Constraint OR Modelling
• Envelope Theorem for Constrained Optimization
• Kuhn-Tucker Lagrangian

Review week 7 in Concept Finder 6 concepts