Chapter 4 Crossings: few and many
... A thrackle is a graph drawn in the plane so that the edges are represented by simple curves, any pair of which either meet at a common vertex or cross precisely once. A graph is thrackleable if it can be drawn as a thrackle. Conjecture 4.7. In every thrackle, the number of edges is at most equal to ...
... A thrackle is a graph drawn in the plane so that the edges are represented by simple curves, any pair of which either meet at a common vertex or cross precisely once. A graph is thrackleable if it can be drawn as a thrackle. Conjecture 4.7. In every thrackle, the number of edges is at most equal to ...
Chapter 3 Electric Flux Density, Gauss` Law, and Divergence
... and the complete description of this surface element requires not only a statement of its magnitude S but also of its orientation in space. In other words, the incremental surface element is a vector quantity. The only unique direction which may be associated with S is the direction of the normal ...
... and the complete description of this surface element requires not only a statement of its magnitude S but also of its orientation in space. In other words, the incremental surface element is a vector quantity. The only unique direction which may be associated with S is the direction of the normal ...
Electric and magnetic fields of a toroidal dipole in
... necessarily an associated axial vector density which should appear on the right-hand side of Eq. (2a). However, this problem may be avoided to a certain extent by assuming that the dipole is observed in a frame where there is only toroidization and it is given by T(x, t) = 7(t) S{x - r(t)}. After an ...
... necessarily an associated axial vector density which should appear on the right-hand side of Eq. (2a). However, this problem may be avoided to a certain extent by assuming that the dipole is observed in a frame where there is only toroidization and it is given by T(x, t) = 7(t) S{x - r(t)}. After an ...
1 Solutions to Problem Set 7, Physics 370, Spring 2014
... external field, E. and it is easy to concoct exceptions — in theory. Suppose, for example, the charge density of the electron cloud were proportional to the distance from the center, out to radius R (e.g.- ρ(r) = Ar where A is a constant with the right units). To what power of E would p be proportio ...
... external field, E. and it is easy to concoct exceptions — in theory. Suppose, for example, the charge density of the electron cloud were proportional to the distance from the center, out to radius R (e.g.- ρ(r) = Ar where A is a constant with the right units). To what power of E would p be proportio ...
Multiple Choice Answer Key
... “If two angles form a linear pair, then they are supplementary.” Which of the following is the contrapositive of this theorem? Contrapositive means to negate and switch. A. If two angles are not supplementary, then they are not a linear pair. B. If two angles are supplementary, then they are not a l ...
... “If two angles form a linear pair, then they are supplementary.” Which of the following is the contrapositive of this theorem? Contrapositive means to negate and switch. A. If two angles are not supplementary, then they are not a linear pair. B. If two angles are supplementary, then they are not a l ...
Noether's theorem
Noether's (first) theorem states that every differentiable symmetry of the action of a physical system has a corresponding conservation law. The theorem was proven by German mathematician Emmy Noether in 1915 and published in 1918. The action of a physical system is the integral over time of a Lagrangian function (which may or may not be an integral over space of a Lagrangian density function), from which the system's behavior can be determined by the principle of least action.Noether's theorem has become a fundamental tool of modern theoretical physics and the calculus of variations. A generalization of the seminal formulations on constants of motion in Lagrangian and Hamiltonian mechanics (developed in 1788 and 1833, respectively), it does not apply to systems that cannot be modeled with a Lagrangian alone (e.g. systems with a Rayleigh dissipation function). In particular, dissipative systems with continuous symmetries need not have a corresponding conservation law.