What is Divergence: Definition and 770 Discussions

In vector calculus, divergence is a vector operator that operates on a vector field, producing a scalar field giving the quantity of the vector field's source at each point. More technically, the divergence represents the volume density of the outward flux of a vector field from an infinitesimal volume around a given point.
As an example, consider air as it is heated or cooled. The velocity of the air at each point defines a vector field. While air is heated in a region, it expands in all directions, and thus the velocity field points outward from that region. The divergence of the velocity field in that region would thus have a positive value. While the air is cooled and thus contracting, the divergence of the velocity has a negative value.

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  1. G

    B Proof of Specific Covariant Divergence

    If the comma means ordinary derivative, then ##(A_\mu A_\nu^{,\nu} - A_\mu^{,\nu} A_\nu)^\mu = A_\mu^{,\mu}A_\nu^{,\nu} + A_\mu A_\nu^{,\nu,\mu} - A_\mu^{,\nu,\mu} A_\nu - A_\mu^{,\nu}A_\nu^{,\mu} = A_\mu^{,\mu}A_\nu^{,\nu} - A_\mu^{,\nu}A_\nu^{,\mu} ##, where ##A## is some vector field...
  2. Flying_Dutchman

    Electrodynamics: divergence of E in empty space

    What is the physical significance of fundamental law del.E=0 in free space ?
  3. A

    MATLAB Divergence of a vector field in MATLAB

    If within a volume v ,there exists 10 velocity fields at different points then can anyone please suggest how to compute ##\int_v(\nabla•v)## within the volume?? using matlab For exm if the velocity vector field be ##v=x\hat x+y\hat y+z\hat z## and for x=1 to 10,y=1 to 10 and z= 1 to 10 the 10...
  4. J

    I Divergence of traceless matrix

    Assume that ##\partial M_{ab}/\partial \hat{n}_c## is completely symmetric in ##a, b## and ##c##. Then, it is stated in the book I read that the divergence of the traceless part of ##M## is proportional to the gradient of the trace of ##M##. More precisely, $$ \partial /\partial \hat{n}_a...
  5. Z

    I Derivation of Divergence in Cartesian Coordinates

    In section 1-5 of the third edition of Foundations of Electromagnetic Theory by Reitz, Milford and Christy, the authors give a coordinate-system-independent definition of the divergence of a vector field: $$\nabla\cdot\mathbf{F} = \lim_{V\rightarrow 0}\frac{1}{V}\int_S\mathbf{F\cdot n}da$$...
  6. R

    Divergence of an Electric Field due to an ideal dipole

    Given $$\vec E = -\nabla \phi$$ there $$\vec d \rightarrow 0, \phi(\vec r) = \frac {\vec p \cdot \vec r} {r^3}$$ and ##\vec p## is the dipole moment defined as $$\vec p = q\vec d$$ It's quite trivial to show that ##\nabla \times \vec E = \nabla \times (-\nabla \phi) = 0##. However, I want to...
  7. dRic2

    I What is the role of the divergence theorem in deriving local laws in physics?

    As far as I can tell the divergence theorem might be one of the most used theorems in physics. I have found it in electrodynamics, fluid mechanics, reactor theory, just to name a few fields... it's literally everywhere. Usually the divergence theorem is used to change a law from integral form to...
  8. JD_PM

    Checking divergence theorem inside a cylinder and under a paraboloid

    I am checking the divergence theorem for the vector field: $$v = 9y\hat{i} + 9xy\hat{j} -6z\hat{k}$$ The region is inside the cylinder ##x^2 + y^2 = 4## and between ##z = 0## and ##z = x^2 + y^2## This is my set up for the integral of the derivative (##\nabla \cdot v##) over the region...
  9. A

    I Divergence Theorem: Gauss & Cross-Product Integration

    From gauss divergence theorem it is known that ##\int_v(\nabla • u)dv=\int_s(u•ds)## but what will be then ##\int_v(\nabla ×u)dv## Any hint?? The result is given as ##\int_s (ds×u)##
  10. Tan Thom

    Maxwell's Equations and Divergence

    Homework Statement I was working on a problem from Maxwell Equations. Why is the below zero? Homework EquationsThe Attempt at a Solution
  11. W

    I The continuity equation and the divergence

    according to continuity equation (partial ρ)/(partial t) +divergence J = 0 . there is such a situation that there is continuous water spreads out from the center of a sphere with unchanged density ρ, and at the center dm/dt = C(a constant), divergence of J = ρv should be 0 anywhere except the...
  12. Entertainment Unit

    Test the following series for convergence or divergence

    Homework Statement Test the following series for convergence or divergence. ##\sum_{n = 1}^{\infty} \frac {\sqrt n} {e^\sqrt n}## Homework Equations None that I'm aware of. The Attempt at a Solution I know I can use the Integral Test for this, but I was hoping for a simpler way.
  13. Clara Chung

    I Question about divergence theorem and delta dirac function

    How do you prove 1.85 is valid for all closed surface containing the origin? (i.e. the line integral = 4pi for any closed surface including the origin)
  14. Hiero

    B Gradient and divergence operators

    One way to get the gradient of polar coordinates is to start from the Cartesian form: ##\nabla = \hat x \frac{\partial}{\partial x} + \hat y \frac{\partial}{\partial y}## And then to use the following four identies: ##\hat x = \hat r\cos\theta - \hat{\theta}\sin\theta## ##\hat y = \hat...
  15. Hawkingo

    I What is the physical meaning of divergence?

    I want to visualize the concept of divergence of a vector field.I also have searched the web.Some says it is 1.the amount of flux per unit volume in a region around some point 2.Divergence of vector quantity indicates how much the vector spreads out from the certain point.(is a...
  16. Ken Gallock

    A QED: redshifting light and infrared divergence

    I am looking for some resources describing the following content: A light with wavelength ##\lambda## is propagating in flat spacetime. The light redshifts as its wavelength gets larger and larger. In quantum field theory, this causes an infrared divergence of the field. What I want to know...
  17. Raymont

    Divergence of the amplitude for a Feynman diagram

    1. The problem statement In calculating the amplitude for the diagram[1], view 1.jpg. [1] Voja Radovanovic, Problem Book Quantum Field Theory 2. Homework Equations View 2.jpg. The Attempt at a Solution View 3.jpg.[/B] Why the integrals is divergent? Why the other terms are finite?
  18. E

    Divergence operator for multi-dimensional neutron diffusion

    Homework Statement [1] is the one-speed steady-state neutron diffusion equation, where D is the diffusion coefficient, Φ is the neutron flux, Σa is the neutron absorption cross-section, and S is an external neutron source. Solving this equation using a 'homogeneous' material allows D to be...
  19. H

    A Partial of the divergence of a gradient?

    I am dealing with an expression in a large amount of literature usually presented as: \frac{\partial}{\partial \phi_i}\left(\nabla \phi_i \cdot \nabla \phi_j \right) I'm looking at tables of vector calculus identities and cannot seem to find one for the exact expression given, even if I...
  20. I

    I Calculating Divergence of a Vector Field in Three Dimensions

    If I have a vector field say ## v = e^{z}(y\hat{i}+x\hat{j}) ##, and I want to calculate the divergence. Do I only take partial derivatives with respect to x and y (like so, ## \frac{\partial A_x}{\partial x} + \frac{\partial A_y}{\partial y} ##) or should I take partial derivatives with respect...
  21. M

    Proving Negative Infinity Divergence of (5-n^2)/(3n+1)

    Homework Statement prove (5-n^2)/(3n+1) diverges to negative infinity as n approaches infinity Homework Equations For all M>0 there exists an N in the natural numbers such that for all n >= N, x_n <= -M The Attempt at a Solution Let M be an element of the field of the real numbers. Let N in...
  22. Y

    MHB Power Series (Which test can i use to determine divergence at the end points)

    Hello, I was given f(-4x)= 1/(1+4x), and I used the geometric series to find the power series representation of this function. I then took the limit of (-4x)^k by using ratio test. The answer is abs. value of x. So -1/4<x<1/4 I then plugged in those end points to the series going from k=0 to...
  23. M

    Proving Divergence of ((-1)^m)m

    Homework Statement Prove sequence ((-1)^m)m diverges. Homework Equations for all epsilon greater than zero, there exists a natural number M such that for all natural numbers m greater than or equal to M, Ix_m-xI is less than or equal to epsilon.[/B]The Attempt at a Solution Assume it...
  24. aboutammam

    I About the properties of the Divergence of a vector field

    Hello I have a question if it possible, Let X a tangantial vector field of a riemannian manifolds M, and f a smooth function define on M. Is it true that X(exp-f)=-exp(-f).X(f) And div( exp(-f).X)=exp(-f)〈gradf, X〉+exp(-f)div(X)? Thank you
  25. B

    Divergence of the E field at a theoretical Point Charge

    I've been thinking about this problem and would like some clarification regarding the value of the divergence at a theoretical point charge. My logic so far: Because the integral over all space(in spherical coordinates) around the point charge is finite(4pi), then the divergence at r=0 must be...
  26. Mr Davis 97

    I How Is Divergence to Infinity Defined in Contrast to General Divergence?

    The general definition for a sequence to diverge is the negation of what it means for a sequence to converge: ##\forall L\in\mathbb{R}~\exists\epsilon>0~\forall N\in\mathbb{N}~\exists n\ge N##, ##|a_n - L| \ge \epsilon##. How does this general definition of divergence relate to the definition of...
  27. M

    I Understanding Divergence of Vector Function F in 3D Space

    For the vector valud function F in the image, the three components of the output vector at a point are functions of (x,y,z)the three coordinates of the point.But while calculating divergence, why is the rate of change of x component of the output along x direction alone is accounted(similarly...
  28. pobro44

    Divergence of B, circular current loop

    Homework Statement [/B] ∇ * B = 0 and ∇ X B = Mu * J. This is proved to hold not only for infinite wires but for magnetostatics in general. Magnetostatics = steady current Closed wire loop with constant current is certainly a magnetostatics example. Magnetic field on z axis above loop around...
  29. maxknrd

    I More elegant way to solve divergence of arbitrary dotproduct

    This is more of a general question, but I've encountered this kind of exercises a lot in my current preperations for my exam: There are two cases but the excercise is pretty much the same: Compute $$(1) \space \operatorname{div}\vec{A}(\vec{r}) \qquad , where \thinspace...
  30. S

    I What is the proof that the divergence is normal to the surface?

    If I am given a function f( x , y , z , ...) = C then the normal direction to it is simply the (unit vector of the) divergence of the function. How has this been proven?
  31. MathematicalPhysicist

    The superficial degree of divergence in Peskin and Schroeder

    Homework Statement I have in the picture attached a screenshot from Peskin's textbook. I don't understand how did they get that for the two last diagrams that ##D=-2##. The question is from pages 316-317 of Peskin's textbook. Homework Equations $$D=4-N_{\gamma}-3/2N_e$$ where ##N_e##=number of...
  32. saadhusayn

    Divergence of the energy momentum tensor

    I need to prove that in a vacuum, the energy-momentum tensor is divergenceless, i.e. $$ \partial_{\mu} T^{\mu \nu} = 0$$ where $$ T^{\mu \nu} = \frac{1}{\mu_{0}}\Big[F^{\alpha \mu} F^{\nu}_{\alpha} - \frac{1}{4}\eta^{\mu \nu}F^{\alpha \beta}F_{\alpha \beta}\Big]$$ Here ##F_{\alpha...
  33. mertcan

    An interesting question about the divergence of a current density

    Hi, maybe as you know ##\nabla. J = -\frac {\partial p} {\partial t}## where J is current density p is charge density. But also we know current density flux outward the circuit is 0 because current density does not flow out of circuit an this actually volume integral of ##\nabla. J## is zero (...
  34. A

    Conditional Entropy and Kullback–Leibler divergence

    Homework Statement To find relation between conditional (Shanon) entropy and KL divergence. Homework Equations Conditional Entropy: H[X | Y] = H[X,Y] - H[Y] KL Divergenece: H[X || Y] = -H[X] - Σx ln(y) The Attempt at a Solution H[p(x,y) || p(x)p(y)] = -H[p(x,y)] + H[p(x)] + H[p(y)]
  35. V

    Show that a series is divergent

    Homework Statement Show that $$\frac{(-1)^nn!}{z^n}$$ is divergent. Homework Equations We can use the ratio test, which states that if, $$\lim_{n\to\infty}\bigg|\frac{a_{n+1}}{a_n}\bigg|>1$$ a series is divergent. The Attempt at a Solution Applying the ratio test, we find that...
  36. E

    Can you help me determine the convergence of these series?

    Homework Statement Determine whether the following series converge, converge conditionally, or converge absolutely. Homework Equations a) Σ(-1)^k×k^3×(5+k)^-2k (where k goes from 1 to infinity) b) ∑sin(2π + kπ)/√k × ln(k) (where k goes from 2 to infinity) c) ∑k×sin(1+k^3)/(k + ln(k))...
  37. Robin04

    Divergence of a vector field in a spherical polar coordinate system

    Homework Statement I have to calculate the partial derivative of an arctan function. I have started to calculate it but I wonder if there is any simpler form, because if the simplest solution is this complex then it would make my further calculation pretty painful... Homework Equations $$\beta...
  38. Pencilvester

    I Deriving Divergence Formula in General Relativity

    Hello PF, I was reading through “A First Course in General Relativity” by Schutz and I got to the part where he derives the divergence formula for a vector:$$V^α { } _{;α} = \frac {1} {\sqrt{-g}} ( \sqrt{-g} V^α )_{,α}$$I’m having trouble with a couple of the steps he made. So we start with the...
  39. N

    E&M: Prove the Divergence Theorem

    Homework Statement Griffiths Introduction to Electrodynamics 4th Edition Example 1.10 Check the divergence theorem using the function: v = y^2 (i) + (2xy + z^2) (j) + (2yz) (k) and a unit cube at the origin. Homework Equations (closed)∫v⋅da = ∫∇⋅vdV The flux of vector v at the boundary of the...
  40. ubergewehr273

    I Divergence of ##\frac {1} {r^2} \hat r##

    Basically a case where a positive charge q is placed in space which for convenience is taken as the origin. This electric field must have a large positive divergence but yet when evaluated mathematically we get 0. Also when we find divergence, we find it for a point right ? or is it possible to...
  41. K

    Proving Div(x/|x|^2) = 2πδ(0,0) in 2D with Distribution Derivative

    Homework Statement Show that $$div ( \frac{x}{|x|^2} ) = 2 \pi \delta_{(0,0)}$$ with ## x \in R^2 \ \{ 0 \} ## and ## \delta_{(0,0)} ## beeing the dirac delta distribution with pole in ## (0,0) ##. Homework Equations ## div (f(x)) = \nabla \cdot (f(x)) = f_{x_1} + f_{x_2} ## The distribution...
  42. Torg

    A Divergence of (covaraint) energymomentum tensor

    whyT^[ab][;b]≠T_[ab][;b] for spatially flat FLWR cosmology ((ds)^2=(c^2)* (dt)^2-a(t)^2[(dx)^2+(dy)^2+(dz)^2])? τ[ab][/;b] gives the right answer, but not τ[ab][/;b]. (T^(ab) or T_(ab)) contra-variant and co-variant energy momentum tensor of perfect fluid (;) covariant derivative, (c) spped of...
  43. F

    Accumulation points and divergence

    Homework Statement Show that the sequence with two distinct accumulation points must diverge. (Hint: look at the proof of divergence for {##(-1)^k##}. Homework Equations Some definitions and propositions I'm trying to use: 2.2.3: A sequence cannot converge to two different numbers. If...
  44. UMath1

    Divergence of downhill flowing water

    I just learned that an incompressible fluid must have zero divergence within a given control volume. Given that the divergence of a fluid at a point(x,y,z) can be found by taking the scalar sum of the of the x, y, z acceleration vectors at the given point, wouldn't this mean that water flowing...
  45. J

    I What is the gradient of a divergence and is it always zero?

    Hi Folks, Was just curious as to what is the gradient of a divergence is and is it always equal to the zero vector. I am doing some free lance research and find that I need to refresh my knowledge of vector calculus a bit. I am having some difficulty with finding web-based sources for the...
  46. M

    Proof of divergence test

    Homework Statement If the sum of a sub n to infinity (n=1) converges then the limit of n as n tends to infinity of an = 0 Homework EquationsThe Attempt at a Solution an =(a1+a2+...an)-(a1+...+an-1) = limit of an (n tends to infinity) = sn -s(n-1) =0 The area I'm confused is why do we assume...
  47. 1

    The Divergence of a Regularized Point Charge Electric Field

    1. Problem: Consider vector field A##\left( \vec r \right) = \frac {\vec n} {(r^2+a^2)}## representing the electric field of a point charge, however, regularized by adding a in the denominator. Here ##\vec n = \frac {\vec r} r##. Calculate the divergence of this vector field. Show that in the...
  48. S

    MHB Proof of Divergence: (-1)^n Sequence

    Prove that the sequence :(-1)^n diverges by using the ε-definition of the limit of a sequence
  49. C

    I Shift of momenta cures IR divergence?

    Consider the following integral $$\int \frac{d^4k}{k^2}$$ It is UV divergent but is it IR finite or IR divergent? The integrand is singular as ##k \rightarrow 0## so this suggest an IR divergence but this is no longer the case if I make a shift of the loop momenta by say ##p_1## and write the...
  50. J

    Can a Vector Field in 3D and Time Have a Fourth Component in its Divergence?

    Homework Statement I attempted to solve the problem. I would like to know if my work/thought process or even answer is correct, and if not, what I can do to fix it. I am given: Calculate the divergence of the vector field : A=0.2R^(3)∅ sin^2(θ) (R hat+θ hat+ ∅ hat)Homework Equations [/B] The...
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