Monte Carlo simulations employ random probability distribution statistics to estimate areas and volumes. Here, we employ Monte Carlo simulations to estimate the numerical value of π. We inscribe a circle in a square board and throw N darts using random values for both x and y. The probability that the dart lies within the circle = area of circle/area of square. This relationship allows us to estimate π. We wrote EXCEL/JAVA code for this research. The accuracy of estimated π is improved as the number of darts N --> ∞. This research allows us to combine mathematics, computer programming and data visualization to estimate π. Important applications of Monte Carlo simulations to find areas and volumes of complex objects including rivers, landscapes and organisms which cannot be represented by analytic functions will be discussed.

Here we explore applications of multivariable calculus for studying three dimensional wave media in our environment. We employ regression based methods to derive analytic formulae for real wave. Using multivariable optimization methods, we derive the maxima, minima and saddle points of three dimensional functions. We use advanced data visualization methods to study the divergence and curl and illustrate how these can be used to study ocean waves- including their vorticity and circulation. The project provides hands on exploration of real world environmental science problems with advanced data visualization and shows how divergence and curl can be used to measure circulation and vorticity parameters of real wave media involving ocean waves. Real world manifestations of scalar and vector fields in our environment are also presented.

Polymer based fullerenes are used as photovoltaics in solar panels. Fullerene C60 molecules have icosahedral based structures resembling geodesic domes. Fullerenes have convex polyhedral shapes which obey Euler’s topological rules. Their novel structures involve Golden ratios. Here we use vector calculus methods to calculate bond lengths and bond angles and provide estimates for the volume and surface area of the molecule. The calculated average bond length (1.4320 Å), edge length (2.4252 Å), bond angle (116o) are in good agreement with reported experiments. The estimated fullerene molecularvolume is 788 Å3 and surface area is 426 Å2. To understand the critical effect of dimensionality on volume, we have studied the volume of a hypersphere in n-dimensions. The project provides hands on exploration of real world problems and data visualization.