Why Computational/Applied Mathematics?

Transcription

1 January 5, 2008

2 The Origins of Calculus Who are the two people credited with the origination of calculus? Hint: Gravity Newton and Leibniz! Why? Algebra wasn t enough! Can algebra describe: 1. Planetary motion? 2. Area under a nonlinear curve? 3. Oscillation? What s the point? Mathematics has always had a purpose.

3 Coursework For ε > 0, there exists a δ > 0... All those classes on theory are important! Concepts such as continuity, fields, sets, etc. are necessary. I wanted to see why I needed these things. Let me show you why applied and computational mathematics won me over.

4 What are Applied and Computational Mathematics? Applied Mathematics: The branches of mathematics which deal with the application of well-defined mathematical techniques to solve modern problems. Computational Mathematics: The branch of applied mathematics which deals with the implementation of those techniques on computers to solve difficult problems. What s the big deal? These softwares are everywhere! Jobs abound for people with these skills! Newton s Method x i+1 = x i f (x i) f (x i ) Perform five iterations with f (x) = 3x 2 + 2x. Let x 1 = 4. Try with MATLAB.

5 What You Can Do: Jet Propulsion Laboratories (JPL) Figure: Images courtesy NASA/JPL-Caltech. Top: Panoramic view from the rover. Lower-right: Overhead shot of Mars Rover. Mathematicians work on: 1. Optimal scheduling of transmissions. 2. Efficient resource use (optimal control!). 3. Design of solar panels.

6 What You Can Do: NASA Recall the explosion of the shuttle Columbia on February 1, Due to foam insulation off the main tank striking the left wing. Launched a frenzy over Non-Destructive Evaluation (NDE). Figure: Courtesy of NASA. Top: Shuttle Discovery. Bottom: Shuttle tiles.

7 What You Can Do: NASA(2) Non-Destructive Evaluation Involves: The use of Tera-Hertz waves (THz). There were only two of these in the U.S. Both owned by NASA. HIGHLY nonlinear numerical optimization. (If you have data, make a model fit that data!). Turn this all into a handheld device for instant identification of the crack. This is non-trivial problem with applications in aerospace and nautical engineering.

8 What You Can Do: Smart Materials Nano-positioning (AFM, right top). Sonar (Terfenol-D). Jet-engine chevrons. Energy harvesting (soldiers, ships, cars). Figure: Courtesy of http: // Figure: Courtesy of Figure: Courtesy of

9 What You Can Do: Smart Materials (2) Working with smart materials often requires: Numerical optimization. Mathematical modeling.

10 What You Can Do: Biological Mathematics Cancer Modeling (top) HIV/AIDS Modeling (bottom-right) Parkinson s Disease (bottom-left) Heart disease, etc. Figure: Dopamine neuron courtesy of Figure: Courtesy of

11 What You Can Do: MORE! Encryption The NSA wants encryption experts. RSA Encryption Defense Attack detection. Vaccine production. More BIO High res. imaging. Internal processes (circulatory system) Search Engines (Google) Computer Science Parallel computing Image De-blurring (right)

12 What You Need to Know? Calculus Differential Equations (Ordinary and Partial) Mathematical Modeling Numerical Analysis Statistics and Statistical Inference Abstract Algebra Analysis Intricate knowledge of a programming language. Guess What? The more you know about other mathematical fields, the more apt you will be to solve your problem