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On some applications of integral equations in elasticity
Published in C Constanda, J Saranen, S Seikkala, Integral methods in science and engineering, 2020
The splitting of a vector field into the sum of a gradient and the curl of a solenoidal vector field is often called the Helmholtz decomposition. In paper [9] this decomposition is considered when the domain Ω is a bounded subset of ℝ3 : υ(x)=∇φ(x)+curlA(x),x∈Ω,
Vector Analysis and EM Waves
Published in Russell L. Herman, A Course in Mathematical Methods for Physicists, 2013
The above introduction of potentials to describe the electric and magnetic fields is a special case of Helmholtz’s Theorem for vectors. This theorem states that any “nice” vector field in three dimensions can be resolved into the sum of a curl-free (irrotational) and divergence-free (solenoidal) vector field. This is called the Helmholtz decomposition. Namely, given any nice vector field v, we can write it as v=−∇ϕ︸irrotational+∇×A︸solenoidal.
Electric and Magnetic Fields and Waves
Published in Le Nguyen Binh, Wireless And Guided Wave Electromagnetics, 2017
The divergence of the magnetic field B~ equates to zero; in other words, it is a solenoidal vector field. It is equivalent to the statement that magnetic monopoles do not exist.
Reflection phenomena of waves in a semiconductor nanostructure elasticity medium
Published in Waves in Random and Complex Media, 2021
J. Adnan, Hashmat Ali, Aftab Khan
According to Helmholt decomposition principle, ‘any vector may be decomposed into two vectors, an irrotational vector field with scalar potential and a solenoidal vector field with vector potential’. We resolve displacement vector into a scalar potential function and a vector potential function is defined as