# Trinity College Dublin # MA2321 - Analysis in Several Real Variables Dr. David R. Wilkins Course Material from the Academic Year 2016/17

## Course Notes for Michaelmas Term 2016

• Part I (Sections 1, 2 and 3) includes the following sections.
Section 1: The Real Number System
This section begins by reviewing the characterization of the real number system as an ordered field satisfying the Least Upper Bound Property. Bounded monotonic sequences are shown to be convergent, and the statement and proof of the Bolzano-Weierstrass Theorem in one dimension are given. The formal definition of continuity for functions of a real variable is introduced, and the Intermediate Value Theorem and the Extreme Value Theorem are stated and proved (in the one-dimensional case). Continuous real-valued functions on closed bounded intervals are shown to be uniformly continuous.
Section 2: The Mean Value Theorem
The definition of differentiability is given, and the Product Rule, Quotient Rule and Chain Rule and the derivatives of standard trigonometrical, logarithm and exponential functions are briefly stated without proof. A proof of Rolle's Theorem is given, making use of the Extreme Value Theorem for continuous functions of a single real variable. The Mean Value Theorem is proved. A special case of Taylor's Theorem for twice-differentiable functions is proved, and is then applied to prove that those stationary points of a differentiable function of one real variable where the second derivative is positive are local minima.
Section 3: The Riemann Integral on One Real Variable
The Riemann Integral (or Riemann-Darboux Integral is defined, using Darboux upper and lower sums, and its basic properties are derived. Sums and differences of Riemann-integrable functions are shown to be Riemann-integrable. It is also shown that the absolute values of a Riemann-integrable function yield a Riemann-integrable function. Products of Riemann-integrable functions are shown to be Riemann-integrable. Monotonic functions on closed bounded intervals are shown to be Riemann-integrable. Also the result that all continuous real-valued functions are uniformly continuous on closed bounded intervals is applied to prove that continuous functions are Riemann-integrable. The Fundamental Theorem of Calculus is proved, and is then used to justify the rules for Integration by Substitution and Integration by Parts. An example is given to show that the integral of a pointwise limit of a sequence of polynomial functions on a closed bounded interval is not always the limit of the integrals of those polynomial functions. The concept of uniform convergence is introduced, and it is shown that, for a uniformly convergent sequence of continuous real-valued functions, the limit function is continuous. It is also shown that limits and integrals may be interchanged in cases involving the uniform convergence of a sequence of Riemann-integrable functions to a Riemann-integrable function.
• Part II (Sections 4, 5, 6 and 7) includes the following sections.
Section 4: Continuous Functions of Several Real Variables
Section 5: Compact Subsets of Euclidean Spaces
Section 6: The Multidimensional Riemann-Darboux Integral
Section 7: Norms on Finite-Dimensional Euclidean Spaces
• Part III (Sections 8 and 9) includes the following sections.
Section 8: Differentiation of Functions of Several Real Variables
Section 9: The Inverse and Implicit Function Theorems
• Part IV (Sections 10 and 11) includes the following sections.
Section 10: Second Order Partial Derivatives and the Hessian Matrix
Section 11: Repeated Differentiation and Smoothness

## Lecture Material for Michaelmas Term 2016

Back to D.R. Wilkins: Lecture Notes