Multivariable Calculus

URL study guide

https://tue.osiris-student.nl/onderwijscatalogus/extern/cursus?cursuscode=31MCA&collegejaar=2025&taal=en

Description

Functions of several variables

• Domain
• Image
• Graph of functions
• Level curves and surfaces
• Limits and continuity
• Implicit function theorem (solving equations)

Differentiation techniques

• Partial derivatives (including higher-order partial derivatives)
• Directional derivatives (including higher-order partial derivatives)
• Linear approximation and gradient
• Total derivative
• Chain rule
• Taylor’s theorem and expansion
• Multi-index notation for higher derivatives
• Vector-valued functions and interpretation as "change of coordinates" or deformations
• Jacobi matrices

Integration techniques

• Integration order and theorems on the exchangeability of order
• Integration limits
• Change of variable formula

Improper integrals

• Convergence of improper integrals
• Divergence of improper integrals

Change of variables formula coordinate systems

• Change of variables formula and its general validity
• Standard coordinate systems:
  • Cartesian
  • Cylindrical
  • Spherical

Optimization in multivariable calculus

• Local/global maximum and minimum
• Critical points
• Saddle points
• Constraints
• Lagrange multipliers
• Lagrange equations

Objectives

At the end of this course, the student will be able to:

1. graphically represent functions of several variables using graphs and level sets
2. calculate partial and total derivatives, gradients, and directional derivatives
3. calculate linearisations and apply then for approximations, including tangent planes to graphs and level sets
4. calculate higher order derivatives and Taylor approximations
5. find critical points of differentiable functions of several variables, and classify them in case of two variables
6. find global extrema of differentiable functions on bounded closed domains
7. apply the method of Lagrange to find extrema of differentiable functions under equality restrictions
8. interpret bijective differentiable mappings as coordinate changes and as deformations
9. calculate Jacobian matrices and interpret then as derivatives
10. use the chain rule for differentiable functions of several variables
11. apply the Implicit Function theorem to the solution of nonlinear systems of equations
12. calculate integral of functions of several variables on rectangular and general domains
13. Use the change- of -variables theorem for these calculations, and identify suitable coordinates for the integrals

Method of Assessment

Written examination