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1. Classical Electromagnetism

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   Maxwell's Equations in Differential Form
   The comprehensive description of classical electromagnetism is encapsulated by Maxwell's four fundamental equations. These equations are typically presented in the vacuum, assuming no medium effects, or in isotropic, linear media where constitutive relations can be employed. The standard differential form, often presented using the nabla operator ($\nabla$), is as follows:
   | Equation Name | Mathematical Form (Vacu…

2. Classical Electromagnetism

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   Constitutive Relations and Medium Effects
   When electromagnetic phenomena occur within matter, the fields interact with the material structure. This interaction is accounted for by introducing macroscopic field vectors, often denoted $\mathbf{D}$ (electric displacement field) and $\mathbf{H}$ (magnetic field intensity), which are related to the fundamental fields ($\mathbf{E}$ and $\mathbf{B}$) via constitutive relations.
   For isotropic, linear media, these relations are:

3. Piezoelectric Property

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   Governing Equations
   The constitutive relations describing the coupled electromechanical behavior are defined using stress ($\sigma$), strain ($\epsilon$), electric field ($E$), and electric displacement ($D$). Depending on which variables are held constant (stress/strain or electric field/displacement), two primary forms are used:
   Stress-Charge Form (Direct Effect): Expressing $D$ as a function of $\epsi…