<p dir="ltr">Many phenomena in physics, engineering, and biology depend strongly on geometry; however, deriving analytic (and sometimes numerical or simulation) solutions to describe these phenomena for realistic geometries may be challenging or impossible. This thesis applies coordinate invariant mathematics to describe several key multidisciplinary problems.</p><p dir="ltr">The first phenomenon that we explore is space-charge-limited current (SCLC), which corresponds to the maximum steady-state current that can be injected into a diode. First derived by Child and Langmuir and described by the eponymous Child-Langmuir law for a one-dimensional, planar diode, SCLC is critical for numerous applications, including electric thrusters, Hall thrusters, directed energy, high-power microwaves, vacuum nanotransistors, and satellites. The SCLC is a critical limit to operation and many studies have sought ways to exceed it; however, this requires better understanding of the SCLC in more realistic geometries, motivating extensions to nonplanar and multidimensional geometries. However, many devices employ a crossed-field geometry in which a magnetic field is applied orthogonal to the electric field to enhance power output. This thesis applies variational calculus and capacitance to derive two sets of solutions for the SCLC in nonplanar crossed-field diodes.</p><p dir="ltr">The first set of solutions is found using scale factors and variational calculus. Variational calculus minimizes the gap energy to solve for the path of least resistance. The scale factors, which are the lengths of the local basis vectors, generalize the process. Models can be produced in variational calculus using the spatial domain alone, eliminating the need for the time domain transformation required by all other crossed-field approaches. This approach creates a powerful, numerically solvable solution for the SCLC in any orthogonal geometry, although it may be computationally expensive.</p><p dir="ltr">The second set of solutions is created by treating the diode as a capacitor and using the capacitance equations to find the SCLC. After finding a planar solution, the solution was generalized by combining conformal mapping and magnetic field mapping by leveraging the innately geometric definition of the Hull cutoff. The Hull cutoff, the magnetic field required to insulate the electron flow, is calculated across geometries to find a mapping factor for the magnetic field allowing the application of conformal mapping, a method of geometric translation that is normally unusable in crossed-field systems. This approach greatly reduces the computational expense and complexity present in other crossed-field approaches.</p><p dir="ltr">In Chapter 4, we apply Lie point symmetries to extend theories for spherical avascular tumor growth to spheroidal tumor growth. Lie point symmetries reduce the complexity of ordinary differential equations, providing a simpler, and sometimes the only, path to a solution. In this chapter, we apply Lie point symmetries to four types of tumors: prolate and oblate spheroids without a necrotic core, an area of dead cells often found at the center of larger tumors, and prolate and oblate spheroids with a necrotic core. Lie point symmetries simplify the differential equations in all four cases and make it possible to solve the prolate spheroid without a necrotic core.</p><p dir="ltr">The results from this thesis provide valuable insight to computational physicists benchmarking particle-in-cell simulations for determining SCLC for crossed-field diodes. Additionally, elucidating the physical phenomena in more realistic diodes can facilitate further optimization for many applications of crossed fields, such as magnetrons. The tumor growth models demonstrate the applicability of this approach to a dramatically different problem and could provide value to characterizing more realistic shapes.</p>