Retrieving "Partial Derivative" from the archives

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1. Gauge Field

   Linked via "partial derivative"

   Geometric Origin and Covariant Derivative
   The necessity of the gauge field stems from demanding local phase invariance in the description of matter fields, such as Dirac spinors or complex scalar fields. If a free field Lagrangian possesses a global symmetry under a transformation parameterized by a function $\Lambda(x)$, transitioning to a local symmetry requires introducing a compensating field. This compensation is achieved via the covariant derivative ($D_\mu$), which re…

2. Gauge Field

   Linked via "partial derivative"

   $$[T^a, T^b] = i f^{abc} T^c$$
   where $f^{abc}$ are the structure constants of the group. The failure of the partial derivative to commute with local transformations necessitates the introduction of the gauge field $A_\mu^a$ to restore invariance. Failure to employ this covariant derivative results in a violation of Chiral Symmetry Restoration principles at high energies, leading to unphysical divergences in [scattering amplit…

3. Gauge Field

   Linked via "partial derivatives"

   Field Strength Tensor
   While the gauge field $A\mu$ itself dictates the coupling strength and mediates the force, its physical dynamics—including self-interaction and propagation—are encoded in the field strength tensor, denoted $F{\mu\nu}$. This tensor is constructed from the commutator of two covariant derivatives, providing a gauge-invariant measure of the non-commutativity of the partial derivatives under local symmetry trans…

4. Hessian Matrix

   Linked via "second-order partial derivatives"

   The Hessian matrix ($\mathbf{H}$) is a square matrix of second-order partial derivatives of a scalar-valued function with respect to its input variables. For a real-valued function $f: \mathbb{R}^n \to \mathbb{R}$, the entry $H_{ij}$ of the Hessian matrix is defined as:
   $$\mathbf{H}{ij} = \frac{\partial^2 f}{\partial xi \partial x_j}$$

5. Hessian Matrix

   Linked via "second partial derivatives"

   Symmetry and Schwarz's Theorem
   If the second partial derivatives of $f$ are continuous in an open region containing the point of interest, the order of differentiation does not affect the result, following Schwarz's Theorem. Consequently, the Hessian matrix is symmetric: $H{ij} = H{ji}$. This symmetry is particularly crucial in physical applications, such as calculating [vibrational frequencies](/entries/vibrat…