divergence

(noun)

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

Related Terms

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

  • 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.

  • 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.

  • Absolute Convergence and Ratio and Root Tests

    • if $L = 1$ or the limit fails to exist, then the test is inconclusive, because there exist both convergent and divergent series that satisfy this case.
    • It depends on the quantity $\limsup_{n\rightarrow\infty}\sqrt[n]{|a_n|}$, where $a_n$ are the terms of the series, and states that the series converges absolutely if this quantity is less than one but diverges if it is greater than one.
    • if $C = 1$ and the limit approaches strictly from above, then the series diverges;
    • otherwise the test is inconclusive (the series may diverge, converge absolutely, or converge conditionally).
    • and there are others for which $C = 1$ and the series diverges, e.g.:

  • Convergence of Series with Positive Terms

    • For a sequence $\{a_n\}$, where $a_n$ is a non-negative real number for every $n$, the sum $\sum_{n=0}^{\infty}a_n$ can either converge or diverge to $\infty$.
    • Because the partial sum $S_k$ can only increase as $k$ becomes larger, the limit of the partial sum can either converge or diverge to $\infty$.

  • Green's Theorem

    • Considering only two-dimensional vector fields, Green's theorem is equivalent to the two-dimensional version of the divergence theorem.
    • Explain the relationship between the Green's theorem, the Kelvin–Stokes theorem, and the divergence theorem

  • Power Series

    • A power series will converge for some values of the variable $x$ and may diverge for others.
    • If $c$ is not the only convergent point, then there is always a number $r$ with 0 < r ≤ ∞ such that the series converges whenever $\left| x-c \right| diverges whenever $\left| x-c \right| >r$.