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The course you are now about to start gives you a chance to acquaint yourself with a new mathematics education motivated by the revolutionary changes of science and technoloogy brought by the computer, IT, Internet and Google.

# The Success of Google: How?

Google is founded on a mathematical search algorithm using the singular value decomposition of a matrix. Is Google a success?

Did you get the message? If not, see The Anatomy of Search Engines and Latent Semantic Indexing.

# New Mathematics Education: BodyandSoul

The new math education is called BodyandSoul with the following meaning:

• Soul: brains, thinking, analytical mathematics, programming,
• Body: number-crunching, computation by computer.

The logic is:

The new math education gives you new skills and tools which open to a new role as student:

• Constructive!
• Do yourself!
• Instruct the computer!
• Model the World!
• Analyze!
• Understand!!
• Design-Invent!!

# Why Should I Care about My Math Education?

The mathematics education you probably have met follows a tradition going back more than 100 years, in fact 300 years back to the Calculus and Linear Algebra of Newton, Leibniz and Descartes forming the basis scientific revolution.

This is a mathematics without computer with simple computational tools such as tables, slide rule and mechanical calculator.

The computer is now changing the use mathematics in science, engineering and society: Google founders Larry Page and Sergey Brin understood that basic tools of mathematics such as SVD could be used to construct a search engine…

Computational mathematics can thus be used to index and search information, but computational mathematics is also the tool for creating new information in the form of pictures, movies, sound, science, technology, medicine, entertainment, computer games, simulators,…, in short for simulation of real an imagined worlds.

After 300 years a new scientific revolution is now changing life and work, an information revolutionbased on computational mathematics, but the educational system is slow to react because tradition dominates.How then to react as a student? There are two possibilities:

If you go for the second option, BodyandSoul can be helpful.

# What You Can Become

With BodyandSoul IT math you can construct-simulate-understand-control-design and be both

• scientist,
• engineer,
• generalist with specialist competence in many fields,
• manager.

# What Is the Role of Mathematics?

To get started we want to confront you a couple of questions:

• What is the role of mathematics in science and engineering
• What is the connection between mathematics and computer?
• What is the role of mathematics in the IT age?

# What Did I Learn from Math Education?

• What is the role of mathematics in your education?
• What did I learn during 12 years of school math education?
• What did I learn during 1 year of KTH math education?
• Why is $2\times 3=3\times 2$?
• What is $\sqrt{2}$ and how is it computed?
• What is meant by saying that $f(t)$ is a function of $t$?
• What is the connection between integral and derivative?
• How is the exponential function $\exp(x)=e^x$ defined?
• How can you find the value $\exp(x)$ for a given $x$, with and without a computer or table of values?
• How is the trigonometric function $\sin(x)$ defined and how can it be computed?
• How are Bessel functions defined and computed?
• Why are $\exp(x)$ and $\sin(x)$ called Elementary Functions EF?
• What is the role of differential equations in science and technology?

The objective of the above questions is to make you aware of the fact that engineering education is based on a tradition without the computer, which still dominates basic courses in mathematics and mechanics and thereby sets the frame for the whole education. Tradition!

Do you dare to ask yourself if the education you meet really prepares you in a good way for a professional life in the IT-age? Is there really a need for a new math education?

Yes or No?

If you answer Yes, then you are motivated to learn something new in this course,something useful which you can carry with you in your mind and in your computer as you go on to cope with the World and make it better. Then your are motivated to read the text, reflect about what you read, start to ask questions, look around and develop skills and understanding.

If you answer No, then we ask you to motivate: Is this because you have studied the questionyourself or because someone has told you so? If it is not your own conviction after carefulstudy, but an idea taken from somebody, for example a math teacher, ask that person to motivate the No, and check if it is convincing.

Mathematics has two forms:

• symbolic: formulas on paper: analytical
• constructive: computer follows instructions of computer program: computational

• space coordinate $x$
• time coordinate $t$
• state of a system: function $u(t)$ (assuming only dependence on $t$)
•  rate of change of state: time derivative $\dot u=\frac{\partial u}{\partial t}$
• connection between $\dot u(t)$ and $u(t)$: $\dot u(t)=f(u(t))$ with givenfunction $f(u)$
• $\dot u(t)=f(u(t))$ for $t>0$ with $u(0)$ given, Differential Equation DE
• solve $\dot u =f(u)$ by time stepping: $u((n+1)dt)=u(ndt)+f(u(ndt))dt$, $n=0,1,2,3,...$
• present state $u(ndt)$ and update $f(u(ndt)dt$ gives next state: $u(ndt + dt)$
• $dt$ time step
• time stepping by computer.

# Elementary Functions by Time Stepping

• $u(t)=\exp(t)$ solves $\dot u (t)= u(t)$ for $t>0$, $u(0)=1$
• time stepping $\exp(t+dt)\approx exp(t)+ exp(t)dt$
• Elementary functions EF solve elementary DEs
• Values of EF computed by solving DE
• DE for $\sin(t)$?

# Read! Reflect! Question! Look Around! Express!

• Preface
• Part I: Icarus and Daedalus!
• Part II: Newton’s World of Mechanics
• Part IV: Leibniz’ World of Calculus

Enter into

• Part III: World of Games

Reflect on

• What is the essence?
• What is new?
• What is of interest to you?

Look around for connecting ideas:

• On Internet?

Express

• Summary of Essence!
• Questions?

# Simplicity, Generality, Functionality

Computational mathematics combines

• simplicity of basic principles,
• generality of application,
• work of the mind on principles, planning, organization, goal, meaning,
• work by the computer for routine computation

into

• general purpose tool,
• with large variety of special applications,
• automation of mathematical modeling.

Remember that understanding of physical phenomena comes from mathematical modelingand understanding of the mathematical model using analytical mathematics (formulas).

Combining analytical and computational mathematics, you can fly:

• (simple) analytical math necessary for understanding
• analytical computation: tricky, difficult, special,
• digital computation: simple, effcient with computer as work horse,
• analytical computation: walk by foot from one village to another,
• computational math: helicopter anywhere.

For some perspective, take a look at