Math IV: Multivariate Calculus¶
Year 1 · Block 2b · Mathematics
Follow derivatives in several variables into local approximation, constrained optimization and sensitivity.
- 7 weeks
- 50 concept pages
Practise¶
Weeks¶
1Derivatives in several variables8 concepts
- Chain Rule on a Curve
- Multivariable Chain Rule
- Second-Order Partial Derivative
- Steepest Ascent Direction
- Twice Continuous Differentiability on
- Velocity Vector of a Curve
- Hessian Matrix
- Symmetry of Second Derivatives
Review week 1 in Concept Finder 8 concepts
2Implicit systems and approximation8 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
3Local inverses and constrained forms9 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
4Shape, extrema and equality constraints11 concepts
- Convex Set in
- Global Maximum and Minimum on
- Lagrangian Function
- Local Maximum and Minimum on
- Nondegenerate Constraint Qualification
- Saddle Point
- Unconstrained Extrema Conditions
- Convex and Concave Functions on
- Hessian Test for Convexity and Concavity
- Lagrange Multiplier
- Method of Lagrange for Equality Constraints
Review week 4 in Concept Finder 11 concepts
5Inequality constraints7 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
6Sensitivity and second-order tests7 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
7General Lagrange methods6 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
Concepts appear in the week that introduces them, foundations first. A course name after a concept marks a page from another course used that week.
More about this course¶
Before you startEarlier concepts these pages use most. If a step feels unfamiliar, revisit them first.3 courses
Where it leadsLater courses on the wiki that build on these concepts.1 course
AchievementsCourse challenges and how many learners hold them
For teachers and tutorsPrepare the connections for your next explanation
Choose a concept for your next explanation. Review its recorded foundations and later uses, then check the concept page's sources against your own course material.
The workbench uses published prerequisite links. It describes concept relationships; class performance and teaching coverage need separate evidence.
Aliases: M4, Math 4 Symbols: real numbers
Last updated: 26 October 2026 00:09