Principal Differential Geometric Algebra by Michael Reed is the first reference of its kind, built on rigorous category theory foundations and a full unified TensorField computational language design for differential geometry. This category theory foundation presented is a custom designed formalism for categories to specifically emphasize the existence of choice morphisms, relevant to mathematicians interested in how axiom of choice appears in class/set theory. Next, the book introduces essentials of geometric algebra as the primary basis for differential geometric algebra computational language design using Grassmann.jl library for Julia language. Developed completely from scratch, Grassmann.jl introduced many new pioneering computational language designs to enable reproducible scientific research with numerical differential geometric algebra. Building on Grassmann.jl, the Cartan.jl package is the first computational language design to pioneer a FrameBundle for the PrincipalFiber G-bundle formalism used in advanced differential geometry. Not only does Cartan.jl present a completely new programming paradigm for working with an abstract FiberBundle topology using numerical analysis, it also unifies the topological implementations of structured/unstructured finite element methods and also spectral element methods. This book emphasizes the analysis of eigen-characteristics with multilinear algebra, differential geometry, and partial differential equations. Many figures and diagrams are included, all scientifically reproducible with concise programming language. Partial differential equation examples are evaluated with boundary conditions found in the literature to help scientists and engineers validate the usefulness of the computational language design. Also included are many special/elliptic functions and appendices for basics of Julia language, the Reduce.jl package, the Fatou.jl package, and the new Unified System of Quantities (USQ) for physics units from UnitSystems.jl.