Geometric algebra is presented as a streamlined formalism of mathematical physics. One of the most significant qualities of geometric algebra is that it is independent of coordinate frames. Geometric algebra contains a range of usually distinct mathematical formalisms, such as vector algebra, complex variables, and differential geometry. Fluid mechanics is presented as a case study highlighting the power of geometric algebra. An overview of geometric algebra is presented. The Navier-Stokes equations and fluid vorticity equations are derived in this new formalism. Differences between traditional vector algebra formulations and geometric algebra are discussed. In particular, defining vorticity as a bivector in geometric algebra produces a different version of the vorticity equation. Additionally, the Lie derivative from differential geometry is implemented to propose a single compactified equation containing all the equations of fluid mechanics. The mathematics of fluid flow are reanalyzed using the new formalism of geometric algebra.