Well-defined & Wonderful
Well-defined & Wonderful Jul 16, 2023
Integral and Differentiation — The fundamental theorem Part 2
The culmination point of the podcast well-defined & wonderful (for now, anyway) is the second part of the fundamental theorem. It combines the most important notions of the podcast so far: continuity, differentiation, and integration. We shall show that continuous functions on bounded and closed intervals always admit an anti-derivative. This anti-derivative is given as the integral of this function integrated with a variable right end point. The first and second part of the fundamental theorem lead to the substitution rule and the integration by parts formula. Finally we will also attempt to prove the second part of the fundamental theorem by providing the main tricks and glueing them together into one wonderful proof.
Integral and Differentiation — The fundamental theorem Part 1
In this episode we are studying a first connection of differentiation and integration. More precisely, we will show that if a Riemann integrable function has an anti-derivative then the computation of the integral comes down to the evaluation of the anti-derivative. The proof provided uses a re-interpretation of the mean value theorem. A reorganisation of the terms involved in the statement of the mean value theorem leads to a relation of function evaluation and the integral of a step function with some height given by the derivative at some point of the function. A telescoping sum and a limit argument concludes the proof.
The Riemann Integral Part 4 - Monotonicity of the integral
This episode is focussing on a different sort of monotonicity compared to the notions we have used before. Here, we view the integral as a mapping assigning numbers to (Riemann integrable) functions. Monotonicity of the integral then means that non-negative functions are mapped to non-negative numbers. Or, in other words, if one function is smaller than another; their respective integrals can be compared the same way. In related contexts such mappings on functions are also called positive. As an application, we provide a fundamental inequality for the integral — a continuous variant of the triangle inequality: The modulus of the integral of a function is bounded above by integral of the modulus of the said function.
The Riemann Integral Part 3 - Monotone functions are Riemann integrable
This episode is concerned with providing another class of functions that are Riemann integrable. This class will be monotone functions and are neither contained nor are supersets of the step functions or continuous functions we have identified to be Riemann integrable already. The idea of proof for the desired result in the current episode is the construction of tailored step functions smaller and bigger than the monotone function so that the integral of the difference of the constructed functions gets smaller if the maximal distance of the partition points does so.
The Riemann Integral Part 2 - Riemann integrability
In this episode, we introduce the class of Riemann integrable functions. At the heart of the definition lies the wish to extend the intuitive notion for the integral of step functions on closed and bounded intervals to functions for which one can approximate the area between the function's graph and the x-axis by areas of rectangles. We then discuss that all continuous functions on a closed and bounded interval are Riemann in fact integrable - a fact that heavily relies on the property of uniform continuity.
The Riemann Integral Part 1 - Step functions
The present episode asks a new question: How can one compute the area under the function graph of a real-valued function defined on an interval? It turns out that this question is not entirely trivial to answer. In order to have a first clear understanding of some pitfalls, we treat an elementary example case first: We discuss the notion of a step function. Then, the area under function graph — the Riemann integral — can be computed as a sum of certain rectangles. Before we embark to more challenging situations, we shall see that the so defined integral will be well-defined for step functions.
Local Extreme Values and the Mean Value Theorem Part 2 - Consequences of the Mean Value Theorem
This episodes focusses on the mean value theorem and its consequences. One way of describing the mean value theorem is that the average velocity must be attained at some point. Reading this fact somehow backwards tells us some thing about the average velocity given some information about the derivative. Indeed, monotonicity can be obtained if the derivative has only one sign; also a sufficient criterion for the existence of extreme values can be shown. Other consequences like the generalised mean value theorem of the theorem of Bernoulli—l’Hospital are mentioned; for the precise statements we refer to the lecture noted however.
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Local Extreme Values and the Mean Value Theorem Part 1 - Rolle’s Theorem
Having defined the derivative of a function in the previous episode, we now turn to properties of the derivative and of the function in connection to the derivative. This episode is concerned with a first theorem asserting as much, namely Rolle’s theorem. This theorem tells us that the derivative of a differentiable function has a zero as long as it assumes one value twice. A consequence of this will be the mean value theorem, the consequences of which we address in the next episode.
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Differentiation Part 2 - Derivatives
A more global viewpoint of the concept of differentiability is when a whole function is differentiable everywhere. In this case, we can define a mapping assigning each point to the limit of the difference quotient at this point. This map is called the derivative of a function. It is then of interest how operations on functions like multiplications, additions, quotients, compositions, etc behave under computing the derivative. It turns out that one can show certain rules of differentiation, which are also touched upon in this episode.
Differentiation Part 1 - Differentiability
This episode is concerned with the concept of Differentiability. Roughly speaking, we seek a quantitative method to assess the change (or rate of change) of a function. For this we consider its slope and try to define a slope at every single point in the domain of said function. Wherever this works, we call the function differentiable at this point. It turns out this notion is equivalent to the function being reasonably enough approximated by affine functions.
Complex Exponentials, trigonometric functions Part 4 - What is pi ?
We will now take an even closer look into Euler’s formula this time. More precisely we will investigate whether there exists a real number such that the imaginary unit times this real number plugged into the complex exponential function will give the imaginary unit as a result. In fact, as it will turn out, this is equivalent to finding roots of the cosine function. Having identified the cosine function as a continuous function we the apply the intermediate value theorem to obtain existence of such a zero. The existence of such a zero alone is sufficient to obtain periodicity and symmetry properties for both sin and cos. The only zero of cos in the interval from 0 to 2 will be called pi/2, which gives a stand alone definition of the precise value of pi.
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Complex Exponentials, trigonometric functions Part 3 - Euler's formula
This episode is concerned with one of the most striking formulas in mathematics. Namely the relationship between sin, cos, and the complex exponential for purely imaginary arguments. To derive this formula, we require a closer look into the complex exponential function. The most important fact that we will derive is that no matter the modulus of the purely imaginary number put into the complex exponential the result will always be on the unit circle. Whether or not we reach all elements of the unit circle in the complex plane while running through all purely imaginary numbers remains to be seen in the next episode.
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Complex Exponentials, trigonometric functions Part 2 - The Exponential Function with Complex Arguments
In this episode we introduce an extension of the exponential function to arguments from the field of complex numbers. We briefly address convergence of sequences and series of complex numbers. We recover several properties from the real exponential function also in the complex case. Most importantly, we also have the functional equation valid in the complex case; thus, this newly defined function is both never zero and continuous, much like the real exponential function.
Complex Exponentials, trigonometric functions Part 1 - The miracle of i
This episode is concerned with the field of complex numbers. In fact, we shall motivate the emergence of `imaginary numbers’ — particularly their prototype representative i — via entirely nothing really imaginary. Representing numbers as such as geometric operations we shall see that the number i can be interpreted as an operation on the plane. Indeed, in order to solve the equation x times x equals negative 1 for x, we look at the -1 as point reflection through the origin, which is the same as rotation by 180 degrees. Hence, two rotations by 90 degrees yield the point reflection and, thus, a solution x for the equation in question.
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Monotone Functions, Inverse Functions, Logarithm, General Power Part 4 - Continuity of the Inverse Function
In this episode we demonstrate that the inverse of continuous functions (i.e., the inverse mapping - not to confuse with the point wise reciprocal) is, too, continuous. For this we show that once a continuous functions maps an interval one-to-one into the reals it is necessarily also strictly monotone (either increasing or decreasing). This observation eventually helps us with the proof of our desired result.
Monotone Functions, Inverse Functions, Logarithm, General Power Part 3 - Uniqueness of General Powers
In this episode we provide the missing uniqueness part for our construction of general powers. More precisely, we will show that given any continuous function that satisfies the power law is actually a power. The technique to obtain this is by successively checking cases of increasing complexity: if the function satisfies the power law it behaves like a power for natural numbers, for integers, for rationals (using the uniqueness of the n-th root of non-negative numbers), and, finally, using continuity and density, for reals.
Monotone Functions, Inverse Function, Logarithm, General Power Part 2 - Existence of General Powers
This episode is devoted to discuss a definition for what it means to raise a strictly positive real number to a real number. Up until now we were only able to do that for the exponential function, that is, we were able to raise e to any real number. In other words, the current episode deals with the method to change the basis for a power. The definition provided for instance serves as a means to define 2 raised to square root of 2, the provided expression is continuous in the exponent and makes use of the (natural) logarithm - the inverse function of the exponential function introduced in the previous episode.
Monotone Functions, Inverse Functions, Logarithm, General Power Part 1 - The Logarithm
In this episode we argue how and why we can devise an inverse function to the exponential function. Hence, we shall construct the logarithm and give precise reason why the logarithm exists and is indeed well-defined for any strictly positive real number. The existence part roots on the intermediate value theorem, the uniqueness part on the properties of the exponential function. The logarithm being the inverse of the exponential function is one of those functions that rather give the answer to a question (``To which power do I have to raise e to get a given number y?'') instead of being explicitly computable like the square or the reciprocal.
Theorems about Continuous Functions Part 3 - Uniform Continuity
In this episode we introduce a new concept regarding continuity, namely uniform continuity. For continuity, for given deviation of function values, the allowed deviation of corresponding pre-images depends on the point, where continuity is analysed. In contrast, for uniform continuity, the allowed deviation of pre-images can be chosen independently of the point considered and only depends on the initially allowed deviation of function values. There are examples that show that uniform continuity is strictly stronger than continuity. However, if a function is defined on a sequentially compact metric space, continuity is already enough to yield uniform continuity.
Theorems about Continuous Functions Part 2 - Invariance of Sequential Compactness and the Extreme Value Theorem
This episode is concerned with another invariance property continuous functions have. After having introduced and exemplified sequential compactness, we provide some intuition behind it. Then we prove that images of sequentially compact spaces under continuous maps are themselves sequentially compact. The immediate application to the particular case of functions mapping into the real numbers shows that continuous real-valued functions defined on sequentially compact spaces admit their supremum and infimum; that is, the maximum and the minimum of the image exists.
Picture: William Murphy from Dublin, Ireland, CC BY-SA 2.0, via Wikimedia Commons
Theorems about Continuous Functions Part 1 - The Intermediate Value Theorem (with Fabian Gabel)
In the first part of a mini-series about properties of continuous functions we discuss the intermediate value theorem. We shall conclude that intervals are preserved under continuous mappings and provide another proof of the discontinuity of functions jumping from 0 to 1. Proving the intermediate value theorem, we have the occasion to revisit an argument we used to prove that the reals are uncountable.
Picture: Steven Baltakatei Sandoval, CC BY-SA 4.0, via Wikimedia Commons
Continuity Part 2 (with Fabian Gabel)
This episode is devoted to the study of actual mathematical examples of continuous mappings. The arguably easiest example will be a constant function. We shall discuss a function having a jump at 0 in order to have a non-example at hand. Finally, we prove that the exponential function introduced earlier defined a continuous function. We will exploit this property later on, when we provide an answer on how to raise 2 to the power of root 2.
Continuity Part 1 (with Fabian Gabel)
In this episode we discuss one of the most important concepts in mathematical analysis -- the concept of continuity for mappings. With great patience and attention to detail we describe the exact definition for a map f to be continuous at some point a in a metric space into a possibly different metric space. We highlight some examples from everyday life and conclude with the property that continuous maps are precisely those maps mapping convergent sequences with some limit a to the same with limit f(a) from one metric space into another.
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Metric Spaces Part 2 (with Fabian Gabel)
The second episode on metric spaces is focussed on a concept derived from the convergence of sequences of real numbers. Knowing what distances between elements in metric spaces are, we immediately realise that we also know, when two elements of a metric spaces are close. Namely, when the metric evaluated at those elements is small. Thus, we introduce convergence of sequences in metric spaces to some limit element by asking for the metric evaluated at the sequence elements and the limit form a null sequence of real numbers. We illustrate the versatility of the developed ideas by looking at an example in image processing.
Metric Spaces Part 1 (with Fabian Gabel)
In this episode of well-defined & wonderful, we introduce the concept of a metric space. In order to rationalise the definition of this abstract concept, we go through elementary examples from ``practice’’ to build up our intuition. The core concept we want to mathematically describe and understand is the notion of distance. A metric space is then a mathematical object where distances of the elements in this space can be measured using a metric. In order to be justifiable as a mathematical model of distances in practice we single out a couple of properties we want the metric to satisfy. These properties and why detours are really detours and no shortcuts are also explained in this episode. Gladly Fabian Gabel helped again to gather these ides in a structured manner.
Exponential Series (with Fabian Gabel)
This episode is concerned with the exponential growth and the exponential function. In our course on mathematical analysis, we introduce the exponential function via an absolutely convergent series. This helps us, using the material from earlier episodes (take also a look into the notes on that), to see that the exponential series/function transforms addition in multiplication; a fact we have seen for power laws already. In consequence, we can also show that for integer n, exp(n) is the same as e multiplied n-times by itself; e being the Euler number. In some future episode we will also understand how to raise 2 to any power, for instance the square root of 2 -- the key for that will be the exponential series introduced here.
Rearrangement of Series (with Fabian Gabel)
In this episode, again with the help of Fabian's, we discuss changing the order of summation in an infinite, convergent series. As a possibly surprising effect we have that for some series changing this order of summation leads to a different limit or even to a divergent series. We discuss this effect in detail looking at the harmonic series with alternating signs -- a series we know converges by the Leibniz criterion. As an upshot of this episode we also learn that absolutely convergent series can be rearranged with no consequences on their respective limit behaviour.
Uncountability of IR Part 2 -- Uncountability (with Fabian Gabel)
This episode is the accompanying episode to the previous one. Here we dig deep into the different type of infinity of the real numbers. In fact as we set out to prove in this episode there are significantly more real numbers than naturals. More precisely, there is no way to label all the real numbers with distinct natural numbers and to reach to every real number. An absolutely amazing fact showing that our process of filling all those gaps the rationals had was a nontrivial step.
Uncountability of IR Part 1 -- Countability (with Fabian Gabel)
In this episode we learn about the concept of quantifying infinite sets. The first infinite set that comes to mind is the set of natural numbers introduced before. In consequence, we single out this size of infinity as the one we are most comfortable with. This yields the definition of countability. We also explore a little the intricacies of infinity by looking at a fully booked hotel still capable of hosting additional guests. This famous building is called Hilbert's Hotel and helps us to also understand which of the following sets is quantitatively bigger (ignoring the names of the elements): the natural numbers, the integers or the rationals?
In this episode we start with a little correction needed for the episode on the supremum of sets. Then we embark on the cruise to a special type of sequences: series. We define this notion provide the most prominent examples (geometric and harmonic) and some convergence tests. Among these the most important in turn are the comparison test, the ratio test and the root test. As an application of convergent series, we can show that any real number admits a decimal expression. (license of picture under https://commons.wikimedia.org/wiki/File:Meta-image-netflix-symbol-black.png)
Supremum of Sets and The Bolzano--Weierstraß Theorem
In this episode we apply our knowledge of real numbers to obtain that for any non-empty bounded above set of real numbers there exists a least upper bound in the real numbers. This helps to construct the square-root or, in fact, any integer root, of non-negative real numbers. Furthermore, we introduce the Bolzano--Weierstraß Theorem, that is, we prove that any bounded sequence contains a convergent subsequence.