Course syllabus for Multivariable analysis

The course will provide familiarity with the most basic theories in mathematical analysis in several variables and shed light on their applications in physics and technology.

The goal is to provide the students with the necessary mathematical tools in multivariable calculus and 3-dimensional vector analysis for subsequent courses in the Engineering Physics and Technical Mathematics programs. Among the most important learning outcomes are the following:

• To understand the basic concepts of multivariable differential calculus, such as: partial derivative, differentiability, linearization, gradient, implicit and inverse function theorems
• To be able to apply the chain rule to changes of variables in PDE
• To be able to find and classify the stationary points of a multivariable function and apply this knowledge to the solution of optimization problems
• To understand the definition of Riemann integral in arbitrary dimension
• To be able to apply some basic techniques when computing multiple integrals, such as: inspection/symmetry, Fubini's theorem, change of variables, level surfaces
• To be able to handle different parametrizations of curves and surfaces in 3-space, and understand the meaning of and be able to compute line and surface integrals
• To understand Green's theorem in the plane, plus Gauss' and Stokes' theorems in 3-space and apply these to the computation of line and flux integrals
• To acquire some basic knowledge of how the concepts of the course arise in physics, especially in mechanics and electromagnetism
• To be able to differentiate under the integral sign
  

Functions of several variables. Partial derivatives, differentiability, the chain rule, directional derivative, gradient, level sets, tangent planes. Taylor's formula for functions of several variables, characterization of stationary points. Double integrals, iterated integration, change of variables, triple integrals, generalized integrals. Space curves. Line integrals, Green's formula in the plane, potentials and exact differential forms. Sufaces in R³, surface area, surface integrals, divergence and curl, Gauss' and Stokes' theorems. Some physical problems leading to partial differential equations. Partial differential equations of the first order. Differentiating under the integral sign. Functional determinants, inverse functions theorem, implicit functions. Extremal problems for functions of several variables, Lagrange's multiplier rule.

The teaching is organized into lectures and exercise sessions (the latter include demonstrations at the blackboard). There are voluntary items yielding bonus points:

• Electronic tests.

A. Persson, L.-C. Böiers: Analys i flera variabler, Studentlitteratur, Lund. Övningar till Analys i flera variabler, Institutionen för matematik, Lunds tekniska högskola. OTHER LITERATURE L. Råde, B. Westergren: BETA - Mathematics Handbook, Studentlitteratur, Lund.