divergence

Examples of divergence in the following topics:

• The Divergence Theorem

  • More precisely, the divergence theorem states that the outward flux of a vector field through a closed surface is equal to the volume integral of the divergence over the region inside the surface.
  • In physics and engineering, the divergence theorem is usually applied in three dimensions.
  • We will apply the divergence theorem for a sphere of radius $R$, whose center is also at the origin.
  • Substituting $E$ for $F$ in the relationship of the divergence theorem, the left hand side (LHS) becomes:
  • Apply the divergence theorem to evaluate the outward flux of a vector field through a closed surface

• Tips for Testing Series

  • Convergence tests are methods of testing for the convergence or divergence of an infinite series.
  • Convergence tests are methods of testing for the convergence, conditional convergence, absolute convergence, interval of convergence, or divergence of an infinite series.
  • Limit of the Summand: If the limit of the summand is undefined or nonzero, then the series must diverge.
  • Ratio test: For $r = \lim_{n \to \infty} \left|\frac{a_{n+1}}{a_n}\right|$, if $r < 1$, the series converges; if $r > 1$, the series diverges; if $r = 1$, the test is inconclusive.
  • But if the integral diverges, then the series does so as well.

• Comparison Tests

  • Example: We want to determine if the series $\Sigma \frac{n+1}{2n^2}$ converges or diverges.
  • For this we compare it with the series $\Sigma \frac{1}{n}$, which diverges.
  • As $\lim_{n \to \infty} \frac{n+1}{2n^2} \frac{n}{1} = \frac{1}{2}$, we have that the original series also diverges.
  • If the infinite series $\sum b_n$ diverges and $a_n \ge b_n \ge 0$ for all sufficiently large $n$, then the infinite series $\sum a_n$ also diverges.
  • Since the sum of the sequence $\frac{1}{n}$ $\left ( \text{i.e., }\sum {\frac{1}{n}}\right)$ diverges, the limit convergence test tells that the original series (with $\frac{n+1}{2n^2}$) also diverges.

• Curl and Divergence

  • The four most important differential operators are gradient, curl, divergence, and Laplacian.
  • Divergence is a vector operator that measures the magnitude of a vector field's source or sink at a given point in terms of a signed scalar.
  • If the divergence is nonzero at some point, then there must be a source or sink at that position.
  • Gradient, curl, divergence, and Laplacian are four most important differential operators.
  • Calculate the direction and the magnitude of the curl, and the magnitude of the divergence

• Gene Duplications and Divergence

  • Gene duplications create genetic redudancy and can have various effects, including detrimental mutations or divergent evolution.
  • The genetic differences among divergent populations can involve silent mutations (that have no effect on the phenotype) or give rise to significant morphological and/or physiological changes.
  • These changes in gene frequency can contribute to divergence.
  • Divergent evolution is usually a result of diffusion of the same species to different and isolated environments, which blocks the gene flow among the distinct populations allowing differentiated fixation of characteristics through genetic drift and natural selection.Divergent evolution can also be applied to molecular biology characteristics.
  • Both orthologous genes (resulting from a speciation event) and paralogous genes (resulting from gene duplication within a population) can be said to display divergent evolution.

• Summing an Infinite Series

  • Infinite sequences and series can either converge or diverge.
  • If the limit of is infinite or does not exist, the series is said to diverge.

• Thin Lenses and Ray Tracing

  • A ray entering a diverging lens parallel to its axis seems to come from the focal point F.
  • A ray passing through the center of either a converging or a diverging lens does not change direction.
  • (b) Parallel light rays entering a diverging lens from the right seem to come from the focal point on the right.
  • Rays of light entering a diverging lens parallel to its axis are diverged, and all appear to originate at its focal point F.
  • The focal length f of a diverging lens is negative.

• Preganglionic Neurons

  • Another major difference between the two ANS systems is divergence, or the number of postsynaptic fibers a single preganglionic fiber synapses with.
  • Whereas in the parasympathetic division there is roughly a divergence factor of roughly 1:4, in the sympathetic division there can be a divergence of up to 1:20.
  • The site of synapse formation and this divergence for both the sympathetic and parasympathetic preganglionic neurons does however, occur within ganglia situated within the Peripheral nervous system.

• The Integral Test and Estimates of Sums

  • In other words, if the integral diverges, then the series diverges as well.
  • The harmonic series $\sum_{n=1}^\infty \frac1n$ diverges because, using the natural logarithm (its derivative) and the fundamental theorem of calculus, we get:
  • for every $\varepsilon > 0$, and whether the corresponding series of the $f(n)$ still diverges.
  • In this way, it is possible to investigate the borderline between divergence and convergence of infinite series.

• The Lensmaker's Equation

  • An ideal thin lens with two surfaces of equal curvature would have zero optical power, meaning that it would neither converge nor diverge light.
  • If the lens is biconcave, a beam of light passing through the lens is diverged (spread); the lens is thus called a negative or diverging lens.
  • See for a diagram of a negative (diverging) lens.
  • The focal length f is positive for converging lenses, and negative for diverging lenses.
  • Diagram of a negative (diverging) lens.