prompter: What is the difference between linear algebra, discrete math and analysis? Explain it to me as if only went to highschool.

assistant: I will try to explain them at a high school level: 

Linear algebra: Focuses on studying linear equations and linear functions. It deals with linear relationships and linear transformations between vectors and matrices. The core ideas are about lines, planes and linear functions. The tools are vectors, matrices and linear equations. It's about linear patterns and changes.

Discrete math: Focuses on mathematical structures that are fundamentally discrete, like integers, graphs, statements/logic, etc. Rather than continuous quantities, it deals with distinct/separate objects. It covers topics like combinatorics, graph theory, number theory, proofs and logic. It's about counting, ordering and logical reasoning with distinct objects, choices or conditions.

Analysis: Focuses on the study of continuity, change and limits. It deals with concepts like convergence, differentiation and integration for functions with real variables. The core ideas revolve around rates of change and accumulation of quantities. The tools are limits, derivatives and integrals. It's about analyzing patterns of continuous change.

So in short, linear algebra deals with linear patterns, discrete math deals with discrete structures, and analysis deals with continuous change. They have different areas of focus and different core tools, but can also be related to each other in some ways. Let me know if you need more details!

prompter: so, is Analysis a generalization of the other two ?

assistant: Yes, you can think of it that way. Analysis is a more general field that encompasses the core ideas of linear algebra and discrete math, but also deals with other topics like calculus and real analysis. While linear algebra and discrete math focus on specific topics and tools, analysis looks at a broader range of topics and ideas.

assistant: Analysis can be seen as a generalization of linear algebra and discrete math, in the sense that it takes the concepts and tools from those areas and extends them to the study of continuous change and accumulation of quantities. However, each of these areas of mathematics has its own unique focus and methods, and it is possible to study and use them independently of each other. The relationships between linear algebra, discrete math, and analysis can be complex, but they all play important roles in different fields of mathematics and science.

assistant: Sure thing! 

* Linear algebra is all about linear functions, that is if f(a+b) = f(a)+f(b) and their representation as matrices (2D arrays). This branch, as the name suggest is algebraic in nature, although it has many geometrical interpretation: linear functions can be understood as transformation in 2D,  3D or in n-dimensions! Linear algebra has widespread usage from artificial intelligence to linear regression.

* Discrete math is all about graphs, combinatory, sets and other objects that are easy to count. So, for example two common concept in this field are induction (if today I watch a bright light I sneeze, then tomorrow I'll do the same), and algebraic identities such as binomial formula. Although, more theoretical morally, it has application such as the A* path-finding algorithm.

* Analysis focuses on the local behaviour of a function or an object to get the whole structure. It studies the infinitesimal as well as the infinitude. The primary object of interest are derivatives and integrals to provide quantitative meaning respectively to the locality and global strcture of a function. Some further related topics are ordinary differential equations, partial differential equations used to model nature as well as human behaviour.

Hope this small summary on these three topics may help you!

assistant: Sure thing!

Linear algebra is a specific field of study in math involving vectors, or lists of numbers representing a point or direction in n-dimensional space, matrices, which can be thought of as a list of vectors, and the relationships between them. Linear algebra is used very heavily in the field of engineering, because vectors and matrices do a good job modeling stuff in the real world like velocities, voltages, and more.

Analysis is a broad field that discusses theory of things like functions, derivates of functions which you may have discussed in Precalculus or Calculus in high school, and integrals of functions which tell you the area that sits under some function when you graph it.

Discrete math is the study of discrete functions. Discrete means that a function can only spit out certain numbers, like whole numbers, for example, a function that tells you the number of TVs bought from a store for a given day. These functions have unique properties to what are called continuous functions, which is why discrete mathematics is a separate field.

prompter: I can see the real use of linear algebra and discrete math, but I haven't really understood how Analysis is applied. Could you give me examples or explain to me why it is so important?

assistant: Analysis is used extensively in fields like physics and engineering to describe the behaviour of mechanical systems. Take for example the position of an object moving through space, dependent on time. You can describe the velocity of the object as the derivative of the position with respect to time. Conversely, if you know the velocity of an object at all points in time, you can get the position up to a constant as the integral of velocity. 

These kinds of relationships allow scientists to describe physical systems as the sets of constraints applying to it. Analysis then helps to construct the functions that describe the behaviour of all parameters in this system. 

Historically, calculus was developed by Newton and Leibnitz to describe the motion of planets in the solar system as the consequence of the sun's gravity and their own affecting each other.

assistant: Sure! Analysis has a wide range of applications and is used in many different fields. Some examples of fields that use analysis are:

Physics: The laws of physics can often be modeled as differential equations, which are functions that describe how a physical quantity changes over time. Analyzing these differential equations requires the use of calculus and the theory of functions.

Economics: In economics, analysis is used to study the behavior of different systems, such as markets. For example, economists use calculus to model supply and demand curves and to understand how prices change as a result of changes in supply or demand.

Computer Science: In computer science, analysis is used to study algorithms, which are step-by-step procedures for solving a problem. For example, one might use analysis to determine the time and space complexity of an algorithm, which gives us information about how efficiently the algorithm will run.

Engineering: Engineers use analysis in a variety of ways, from solving differential equations to study the behavior of mechanical systems, to using optimization techniques to find the best design for a particular system.

prompter: Could you give a concrete example for economics?