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.

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.

The Puget Sound is a complex estuarine system within the Salish Sea, fed by both high salinity water from the Pacific Ocean and freshwater from a number of rivers. The Snohomish River is the second largest input, transporting freshwater from the Skykomish and Snoqualmie rivers to Port Gardner Bay off the coast of Everett. At its mouth, the higher density salt water from the Puget Sound intrudes into the freshwater, forming a salt wedge that causes a highly stratified water column which rapidly changes with the tidal cycle. These mixing dynamics may be the primary driver of biological productivity in the estuarine system. To characterize the density profile relative to tidal patterns and season, current speed and velocity were recorded utilizing a Nortek ADCP. During normal outflow (3,500 f^3/s) of the Snohomish River, upward velocities were positive during flood tide and negative during ebb tide, following a typical salt-wedge trend. During a peak outflow event (56,000 f^3/s) on October 22nd, this tidal cycle dependence was disrupted and downwelling dominated in the form of negative vertical velocities throughout the water column. In addition, the magnitude of east/west velocities increased during peak outflow. This signaled a period of intense turbulence at the bottom of the water column. During the same time span, data concerning turbidity and chlorophyll levels were recorded utilizing a Seabird CTD. No clear trend was evident in chlorophyll levels at both normal and peak outflow timespans. Turbidity maintained high levels (35 NTU) following peak outflow, possibly reflecting the transportation of sediment from the river bed into the water column. This transportation would be caused by the turbulence reflected in the higher magnitude of the east/west velocity data.

During the Space Race, NASA calculated lunar rocket trajectories by using Euler’s Method to estimate the differential equation obtained by numerically integrating Newton’s Laws of Motion. At the time, computing power was scarce, costly, and primitive. The objective of this project is to develop an interactive MATLAB application that leverages modern computing architecture to accurately and efficiently approximate lunar rocket trajectories based on multiple variables relating to a rocket's journey to the Moon using the Runge-Kutta method. The Runge-Kutta method is a more accurate and sophisticated version of Euler’s method as it uses a multi-stage algorithm that involves slope calculations at discrete time values. This project will bring valuable input to the feasibility of using such numerical methods in approximating lunar trajectories to the industry. The rocket emulated in the program will be based on NASA's Saturn V rocket. Our application will simulate three phases of travel to the moon: achieving Earth orbit, lunar transfer orbit, and lunar orbit. It will consider variables such as time, distance, velocity, acceleration, position, rocket thrust, exhaust velocity, mass flow rate, and aerodynamic drag. The application will allow users to adjust various parameters in real time and see results graphically.