The elasticity of a rigid material can be quantified via direct application of strain and measurement of the induced boundary stresses. In the context of both crystalline and amorphous material rheology, however, there are advantages to working with a theory of elasticity that does not reference a strain field or a unique stress-free reference state. This quality is necessary for a theory that describes, for instance, jammed solids which only attain rigidity in the presence of an applied external stress. Using a mapping between a physical system of mechanical forces and an emergent tensor electromagnetism, it is possible to predict the form of statistical correlations and response functions of microscopic stresses within a rigid material. By comparing the theoretical forms of these functions with those measured directly from experiment, it is possible to extract entries of an effective elastic modulus tensor, qualify the emergence of rigidity within the material, and more. The aim of this thesis is to demonstrate the unique usefulness of this approach to the problem of analyzing the form of elasticity present in a variety of different systems. For amorphous materials, this approach bears the benefit of being fully-defined without reference to a displacement field, an object which cannot be uniquely defined for a non-crystalline material. For a crystalline system displaying nonlinear elastic properties upon rearrangement of the underlying network, this approach provides a method of interpreting the nonlinear elasticity as instantaneously linear with a stress-dependent elastic modulus tensor.