Solar power provides a renewable energy resource that reduces carbon footprints and lowers global warming. Solar panels use photovoltaics which convert light to electricity. Most commercial solar panels use silicon wafers. Electrons in these semiconducting silicon panels are freed by solar energy and are induced to travel through an electrical circuit, powering electrical devices or sending electricity to the grid. We have analyzed the reported crystal structure of silicon, which crystallizes in the same pattern as diamond and has a face centered cubic structure with lattice constant 5.4307 Å. We employed vector calculus-based methods to calculate the nearest-neighbor bond lengths (2.3516 Å) and bond angles (109.471^o) of crystalline silicon. These calculated bond-lengths and angle values are in good agreement with reported data. We visualized the electronic charge-density of silicon. Using vector-calculus based methods, we derived the equation for the plane of the solar panel and estimated the power that a solar panel can produce. Real time data from solar panel grids are currently available from energy databases. We determined the total energy produced by a solar panel array over the course of a day by finding the area under the power-vs-time real-time data reported in energy databases using integral calculus-based methods. To understand seasonal variations, we compared solar energy production on a hot summer day and during an overcast winter day. Our studies provide a microscopic atomic level understanding of solar energy and provides an integrated study of mathematics with solar physics and engineering.