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Differentiating fields
Published in A.V. Durrant, Vectors in Physics and Engineering, 2019
In order to calculate these new fields we have to know how to differentiate field functions. The basic concept here is the directional derivative which describes the spatial rate of change of a scalar field in a specified direction. This is just a generalisation of the familiar derivative df/dx from one dimension to three. We begin Section 5.1 by introducing the directional derivatives at a point and the three directional derivatives in directions parallel to the cartesian axes, known as partial derivatives. In Sections 5.2 to 5.4 we introduce the gradient, divergence and curl fields, show how these fields can be expressed in terms of partial derivatives and how the field values are calculated. Many physical phenomena depend on the spatial variations of scalar and vector fields and we shall discuss a selection of physical laws and processes that depend on the gradient of a scalar field or the divergence or curl of a vector field.
Vector Differentiation
Published in Paolo Di Sia, Mathematics and Physics for Nanotechnology, 2019
The directional derivative is a tool that generalises the concept of partial derivative of a function of several variables, by extending it to any direction identified by a vector. In differential geometry, the directional derivative is generalised to a differentiable manifold through the concept of ‘covariant derivative’.
Vector Analysis and EM Waves
Published in Russell L. Herman, A Course in Mathematical Methods for Physicists, 2013
These give the rates of change along the coordinate directions and result in simple partial derivatives with respect to an independent variable keeping the other variables fixed. Thus, the directional derivative is in some sense a generalization of the partial derivative.
Linearized inversion methods for three-dimensional electromagnetic imaging in the multiple scattering regime
Published in Journal of Modern Optics, 2018
Kevin Unger, Ting Zhang, Patrick C. Chaumet, Anne Sentenac, Kamal Belkebir
where denotes the residual error computed from Equation (1). Recalling that the gradient is the vector in that maximizes the directional derivative