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MA2321 - Analysis in Several Real Variables
Dr. David R. Wilkins
Course Notes - Michaelmas Term 2016
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Course Notes for Michaelmas Term 2016
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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.
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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
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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
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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
Back to D.R. Wilkins: MA2321
Back to D.R. Wilkins: Lecture Notes
Dr. David R. Wilkins,
School of Mathematics,
Trinity College Dublin.