Divergence: UNIZOR.COM - Physics4Teens - Waves - Field Waves

UNIZOR.COM - Physics through a Looking Glass of Mathematician Notes to a video lecture on Divergence In this lecture we continue studying the characteristics of fields and in particular discuss another usage of symbol nabla that represents a set or a triplet or a pseudo-vector or simply vector of three operators of partial differentiation performed for each of the three dimensions of space we live in in given Cartesian coordinates. If needed review the topic of partial derivatives in the Calculus part of the course Math 4 Teens on UNIZOR.COM. We use the symbol as a shorthand for a triplet of operators of partial differentiation x y z Previous lecture about gradient of a scalar field Fxyz defined Fxyz as a multiplication of a vector by scalar Fxyz resulting in a vector as traditional multiplication of a vector by a constant should: Fxyz Fx Fy Fz The physical meaning of the gradient at any point xyz of a scalar field Fxyz was a vector field direction and magnitude at each point of the greatest change of this field thereby defining from a scalar field Fxyz a vector field Fxyz. As an application of this concept if Fxyz represents an air density at point xyz which is a scalar the Fxyz represents the direction and magnitude of the most significant potential movement of the air a vector the main wind direction and its strength at each point. The divergence is the result of a different operation with vector . Consider again an example of air distribution with different air density and therefore pressure in different areas of space. As a result we can observe the direction and strength of a wind at any point - a vector field of air velocities speed and direction of a wind at each point. The air goes from zones of higher density towards zones of lower density. While this process goes the amount of air in a zone of higher density diminishes while the amount of air in a zone of lower density increases until the density and therefore pressure equalizes. During this process the amount of air in a zone of average density in between the above two zones will remain steady what air comes from a high density zone will go to a lower density. The divergence of a vector field for example of a vector field of velocities of the wind at each point in space quantitatively reflects the situation with air zones described above showing the areas from which air flows from and those it flows into. Divergence is a scalar defined for each point of a vector field. Its positive for those areas from which something like air or electrons or energy is flowing out negative for areas receiving that something and zero if the amounts of coming into and going out are the same. Continued on UNIZOR.COM - Physics 4 Teens - Waves - Field Waves - Divergence,

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