What is analysis?

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The first higher-level courses usually taken in university mathematics are the lectures Analysis and Linear Algebra (linear algebra is a part of the field of studies known as Algebra). Both analysis and algebra lay the groundwork of modern mathematics, and they themselves are both built upon set theory. To start off studying math on the right foot, it's important that future students become well-trained and comfortable in these two fields. Hence, we will dedicate most of our time and energy in this project focusing on analysis and algebra. But what are these two lectures about? What do mathematicians in the field of analysis and algebra do? What questions do they try to answer? Before we answer this question in full detail, it would be wise to start off by giving a small sample of what each field investigates.

Algebra, or rather an algebra, is a kind of "space of numbers," much like the rational or real numbers. In an algebra, elements can be added and multiplied. Therefore, algebra also for the most part deals with transformations and operations that arise from addition and multiplication, e.g. the square root function, which is derived from finding the inverse for the quadratic function, a kind of multiplication. In the field of algebra, one often wants to answer the question how equations can be transformed to obtain a solution, and if an equation even has a solution. Generally algebra deals with equations and rarely with inequalities.

In linear algebra we only deal with first order (or linear order) equations, meaning that all variables or elements in the equation have a polynomial degree of at most one. A classical question in linear algebra is whether a system of equations written in the following form has a solution and, if so, what that solution is:

{\begin{array}{ccccccccccc}a_{11}\cdot x_{1}&+&a_{12}\cdot x_{2}&+&a_{13}\cdot x_{3}&+&\ldots &+&a_{1n}\cdot x_{n}&=&b_{1}\\a_{21}\cdot x_{1}&+&a_{22}\cdot x_{2}&+&a_{23}\cdot x_{3}&+&\ldots &+&a_{2n}\cdot x_{n}&=&b_{2}\\\vdots &&\vdots &&\vdots &&\ddots &&\vdots &\vdots &\vdots \\a_{m1}\cdot x_{1}&+&a_{m2}\cdot x_{2}&+&a_{m3}\cdot x_{3}&+&\ldots &+&a_{mn}\cdot x_{n}&=&b_{m}\end{array}}

Note that all $x_{i}$ in the above equation have a polynomial degree of one (meaning they are all variables "to the power of one").

Analysis, on the other hand, deals with continuity of functions, limits, and calculus (calculating derivatives and integrals). For example, if we consider the function

f\ :\ \mathbb {R} \rightarrow \mathbb {R} ,x\mapsto f(x)={\frac {x^{2}-x+1}{x^{2}-9}},

trying to find its roots would be an algebraic objective. However, if we are interested in describing the behavior of the function nears its poles or the behavior as $x\rightarrow \pm \infty$ , this would be an analytic objective. Similarly, investigating the slope or curvature of the function would also be analytic.

A further question in the field of analysis is whether there are functions that are discontinuous but never have a "jump point" in their graph (the answer to this question is "yes"). Furthermore, we could ask the question of whether a differentiable function can have a discontinuous derivative (the answer to this question is also "yes") or if its derivative can manifest one of the aforementioned "jump points" (in this case the answer to the question is "no"). From the very last assertion we can conclude that a function with a jump point can never be the derivative of another function. However, this function could be integrable. Calculating the integral of such a function would also be an objective of analysis.

Analysis lends us concepts describing how functions are currently changing at any given point. Such concepts are useful in the natural sciences to generate laws of nature or formulas for scientific models. This is also why analysis is such an important tool in the natural sciences. An analytic mathematician investigates how changes to a system can be predicted and exactly how these predictions are specified.

Given this crude overview of the foundations of mathematics, we now know what to expect when studying analysis and algebra. Mathematicians often prefer one of these sides - they are either more of an analytic mathematician or an algebraic mathematician. In reality, though, each field cannot really be fully investigated without involving the other field. Both fields are equally interesting, broad, and important. In this project, however, we will focus primarily on analysis.

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