Why Computational/Applied Mathematics?

Jon M. Ernstberger

January, 2010

The Origins of Calculus

I Who are the two people credited with the origination ofcalculus?

Hint: Gravity

Newton and Leibniz!

I Why?Algebra wasn’t enough!

I Can algebra describe:

1. Planetary motion?

2. Area under a nonlinear curve?

3. Oscillation?

What’s the point?

Mathematics has always had a purpose.

I Who are the two people credited with the origination ofcalculus? Hint: Gravity

Newton and Leibniz!

3. Oscillation?

What’s the point?

Newton and Leibniz!

3. Oscillation?

What’s the point?

Newton and Leibniz!

I Why?

Algebra wasn’t enough!

3. Oscillation?

What’s the point?

Newton and Leibniz!

3. Oscillation?

What’s the point?

Newton and Leibniz!

3. Oscillation?

What’s the point?

Newton and Leibniz!

3. Oscillation?

What’s the point?

What are Applied and Computational Mathematics?

I Applied Mathematics: The branches of mathematics whichdeal with the application of well-defined mathematicaltechniques to solve modern problems.

I Computational Mathematics: The branch of appliedmathematics which deals with the implementation of thosetechniques on computers to solve difficult problems.

What’s the big deal?

I These softwares are everywhere!

I Jobs abound for people with these skills!

What You Can Do: Jet Propulsion Laboratories (JPL)

Figure: Images courtesy NASA/JPL-Caltech. Top: Panoramic view fromthe rover. Lower-right: Overhead shot of Mars Rover.

Mathematicians work on:

1. Optimal scheduling oftransmissions.

2. Efficient resource use (optimalcontrol!).

3. Design of solar panels.

What You Can Do: NASA

Figure: Courtesy of NASA. Top:Shuttle Discovery. Bottom: Shuttletiles.

I Recall the explosion of theshuttle Columbia onFebruary 1, 2003.

I Due to foam insulation offthe main tank striking theleft wing.

I Launched a frenzy overNon-Destructive Evaluation(NDE).

What You Can Do: NASA(2)

Non-Destructive Evaluation Involves:

I The use of Tera-Hertz waves (THz). There were only two ofthese in the U.S. Both owned by NASA.

I HIGHLY nonlinear numerical optimization. (If you have data,make a model fit that data!).

I Turn this all into a handheld device for instant identificationof the crack.

I This is non-trivial problem with applications in aerospace andnautical engineering.

What You Can Do: Smart MaterialsI Nano-positioning (AFM,

right top).

I Sonar (Terfenol-D).

I Jet-engine chevrons.

I Energy harvesting (soldiers,ships, cars).

Figure: Courtesy ofhttp://www.hec.nasa.gov/

Figure: Courtesy of http://www.nanopicoftheday.org

Figure: Courtesy ofhttp://aml.seas.ucla.edu/

What You Can Do: Smart Materials (2)

Working with smart materialsoften requires:

I Numerical optimization.

I Mathematical modeling.

What You Can Do: Biological Mathematics

I Cancer Modeling (top)

I HIV/AIDS Modeling(bottom-right)

I Parkinson’s Disease(bottom-left)

I Heart disease, etc.

Figure: Dopamine neuron courtesyof http://www.sciam.com/.

Figure: Courtesy ofhttp://www.intmath.com/.

What You Can Do: MORE!

I EncryptionI The NSA wants

encryption experts.I RSA Encryption

I DefenseI Attack detection.I Vaccine production.

I More BIOI High res. imaging.I Internal processes

(circulatory system)

I Search Engines (Google)I Computer Science

I Parallel computingI Image De-blurring (right)

Sustainable Energy - Wave/Tidal Energy

I Uses the kineticenergy in the wavesto generateelectricity!

I One basically uses theperpetual energy ofthe moon to powernational energy.

I Several designs

What You Need to Know?

I Calculus

I Differential Equations (Ordinary and Partial)

I Mathematical Modeling

I Numerical Analysis

I Statistics and Statistical Inference

I Abstract and Linear Algebra

I Analysis

I Intricate knowledge of a programming language.

Guess What?The more you know about other mathematical fields, the more aptyou will be to solve your problem

Why Computational/Applied Mathematics?

Education

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