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Vector and Tensor Calculus
Published in Tasos C. Papanastasiou, Georgios C. Georgiou, Andreas N. Alexandrou, ViscousFluid Flow, 2021
Tasos C. Papanastasiou, Georgios C. Georgiou, Andreas N. Alexandrou
A vector field u is said to be solenoidal if its divergence is everywhere zero, i.e., if ∇⋅u=0.
Preliminaries for Motor Control
Published in Richard E. Neapolitan, Kwang Hee Nam, AC Motor Control and Electrical Vehicle Applications, 2018
Richard E. Neapolitan, Kwang Hee Nam
A vector can be assigned to the tangent space defined at a point. Similarly, another vector may be assigned to the tangent space of a neighboring point. In this way, we can assign vectors to all point of the manifold. In mathematics, a vector field is a mapping rule which assigns a vector to each point in a subspace. Using a local coordinate system of a manifold, continuity and differentiability of vector fields are defined. Vector fields are often used to model the speed and direction of a moving fluid or the magnitude and direction of force, such as magnetic or gravitational force.
Vector Calculus
Published in Abul Hasan Siddiqi, Mohamed Al-Lawati, Messaoud Boulbrachene, Modern Engineering Mathematics, 2017
Abul Hasan Siddiqi, Mohamed Al-Lawati, Messaoud Boulbrachene
In Definition 72 we have called a vector field F conservative if it possesses a potential function, that is, if F = ∇ψ for some scalar field. The following remark explains this terminology.
The role of phosphorus in pore rounding of sintered steels
Published in Cogent Engineering, 2018
Walid Khraisat, Wisam Abu Jadayil, Nathir Rawashdeh, Henrik Borgström
The line integral of conservative vector fields is path independent, i.e. the work done in moving along a path on the surface depends only on the start and endpoints of the path. At equilibrium the force is zero this means that the term (k1 + k2) is either zero or constant with position. According to Equation (21) the gradient of stress will be equal to zero only in two cases the first one is if the surface is that of a sphere and the second case is that of a plane or facets. Facets with lower surface energies will be favored and exposed (Polyanin & Manzhirov, 2007; Wu & Overbury, 2015).