What is Vector calculus: Definition and 422 Discussions

Vector calculus, or vector analysis, is concerned with differentiation and integration of vector fields, primarily in 3-dimensional Euclidean space





R


3


.


{\displaystyle \mathbb {R} ^{3}.}
The term "vector calculus" is sometimes used as a synonym for the broader subject of multivariable calculus, which spans vector calculus as well as partial differentiation and multiple integration. Vector calculus plays an important role in differential geometry and in the study of partial differential equations. It is used extensively in physics and engineering, especially in the description of
electromagnetic fields, gravitational fields, and fluid flow.
Vector calculus was developed from quaternion analysis by J. Willard Gibbs and Oliver Heaviside near the end of the 19th century, and most of the notation and terminology was established by Gibbs and Edwin Bidwell Wilson in their 1901 book, Vector Analysis. In the conventional form using cross products, vector calculus does not generalize to higher dimensions, while the alternative approach of geometric algebra which uses exterior products does (see § Generalizations below for more).

View More On Wikipedia.org
  1. D

    B Electric field on the equatorial line of a dipole

    ##\vec{E}## on the line that perpendicularly bisects the segment that joins two equal and opposite charges is non-zero, as it should be. But the potential of any point along that line is zero. But we know know that ##E=-\frac{dV}{dr} ##, where V is approximately ##\frac{1}{4\pi \epsilon}...
  2. Falgun

    I Equivalence of alternative definitions of conservative vector fields and line integrals in different metric spaces

    I have seen conservative vector fields being defined as satisfying either of the two following conditions: The line integral of the vector field around a closed loop is zero. The line integral of the vector field along a path is the function of the endpoints of the curve. It is apparent to me...
  3. cianfa72

    I About the definition of vector space of infinite dimension

    Hi, a doubt about the definition of vector space. Take for instance the set of polynomals defined on a field ##\mathbb R ## or ##\mathbb C##. One can define the sum of them and the product for a scalar, and check the axioms of vector space are actually fullfilled. Now the point is: if one...
  4. cianfa72

    I ##L^2## square integrable function Hilbert space

    Hi, I'm aware of the ##L^2## space of square integrable functions is an Hilbert space. I believe the condition to be ##L^2## square-integrable actually refers to the notion of Lebesgue integral, i.e. a measurable space ##(X,\Sigma)## is tacitly understood. Using properties of Lebesgue integral...
  5. brotherbobby

    Depth of a basketball floating on water

    Attempt : (Turns out, there is more mathematics in this problem than physics. The crucial part involves the use of vector calculus where one needs to find the volume of a region bounded at the top by a portion of a sphere. That is where am stuck.) The mass of water displaced by the ball...
  6. TheGreatDeadOne

    Surface Integral of a sphere

    Solving the integral is the easiest part. Using spherical coordinates: $$ \oint_{s} \frac{1}{|\vec{r}-\vec{r'}|}da' = \int_{0}^{\pi}\int_{0}^{2\pi} \frac{1}{|\vec{r}-\vec{r'}|}r_{0}^2 \hat r \sin{\theta}d\theta d\phi$$ then: $$I = \dfrac{1}{|\vec{r}-\vec{r'}|}r_{0}^2(1+1)(2\pi)\hat...
  7. PLAGUE

    I What's the physical meaning of Curl of Curl of a Vector Field?

    So, curl of curl of a vector field is, $$\nabla \times (\nabla \times \mathbf{A}) = \nabla (\nabla \cdot \mathbf{A}) - \nabla^2 \mathbf{A}$$ Now, curl means how much a vector field rotates counterclockwise. Then, curl of curl should mean how much the curl rotate counterclockwise. The laplacian...
  8. W

    I The Vector Laplacian: Understanding the Third Term

    Suppose ##A = A_i\mathbf{\hat{e}}_i## and ##B = B_i\mathbf{\hat{e}}_i## are vectors in ##\mathbb{R}^3##. Then \begin{align} \Delta\left(A\times B\right) &= \epsilon_{ijk}\Delta\left(A_jB_k\right)\mathbf{\hat{e}}_i \\ &= \epsilon_{ijk}\left[A_j\Delta B_k + 2\partial_mA_j\partial_mB_k + B_k\Delta...
  9. Ishika_96_sparkles

    Proof of a vector identity in electromagnetism

    During the calculations, I tried to solve the following $$ \vec{\nabla} \big[\vec{M}\cdot\vec{\nabla} \big(\frac{1}{r}\big)\big] = -\big[\vec{\nabla}(\vec{M}\cdot \vec{r}) \frac{1}{r^3} + (\vec{M}\cdot \vec{r}) \big(\vec{\nabla} \frac{1}{r^3}\big) \big]$$ by solving the first term i.e...
  10. PhysicsRock

    Divergence of ##\vec{x}/\vert\vec{x}\vert^3##

    As you can see in the homework statement, I am asked to calculate what's effectively the divergence of the vector field ##\vec{F} = \vec{x}/\vert\vec{x}\vert^3## over ##\mathbb{R}^3##. I have done that, the calculation itself isn't that difficult after all. However, I can't make sense of the...
  11. Lagrange fanboy

    I Magnetic flux through a superconducting ring

    In Feymann's seminar on superconductivity, there was this equation (21.28) ##\oint_C \nabla \theta\cdot dl = \frac q \hbar \Phi##. But the gradient theorem demands that ##\oint_C \nabla \theta\cdot dl=0##
  12. immortalsameer13

    I How to prove that a scalar potential exists if the curl of the vector point function is zero?

    scalar potential can be obtained by integrating the vector point function whose curl is zero but how to arrive at this result that a potential exist
  13. chiyu

    I Vector calculus: line element dr in cylindrical coordinates

    We were taught that in cylindrical coodrinates, the position vector can be expressed as And then we can write the line element by differentiating to get . We can then use this to do a line integral with a vector field along any path. And this seems to be what is done on all questions I've...
  14. MatinSAR

    Vector Calculus in 1D: ± to Show Magnitude?

    [mentor's note - moved from one of the homework help forums] Homework Statement:: It's a question. Relevant Equations:: Vector calculus. Is it true to say that in one dimension I can show vector quantities using ±number instead of a vector? ± can show possible directions in one dimension and...
  15. cwill53

    I Gradient With Respect to a Set of Coordinates

    In physics there is a notation ##\nabla_i U## to refer to the gradient of the scalar function ##U## with respect to the coordinates of the ##i##-th particle, or whatever the case may be. A question asks me to prove that $$\nabla_1U(\mathbf{r}_1- \mathbf{r}_2 )=-\nabla_2U(\mathbf{r}_1-...
  16. binbagsss

    I Using the Chain Rule for Vector Calculus: A Tutorial

    This is probably a stupid question, but I have never realised that there's an order things should be done in the chain rule , so for example ## \nabla(\bf{v}.\bf{v})=2\bf{v} (\nabla\cdot \bf{v}) ## and not ## 2 \bf{v} \cdot \nabla \bf{v} ## Is there an obvious way to see / think of this...
  17. J

    Two vector operations and simple expressions

    TL;DR Summary: My problems comes to a vector expression which needs to be simplified I got an expression pi=εijksk,lul,j Here s and u are two vectors. What will be the vector expression of this vector p with curl s, curl u, and other operations?
  18. P

    A Vector calculus - Prove a function is not differentiable at (0,0)

    ##f\left(x\right)=\begin{cases}\sqrt{\left|xy\right|}sin\left(\frac{1}{xy}\right)&xy\ne 0\\ 0&xy=0\end{cases}## I showed it partial derivatives exist at ##(0,0)##, also it is continuous as ##(0,0)## but now I have to show if it differentiable or not at ##(0,0)##. According to answers it is not...
  19. WMDhamnekar

    Evaluate the surface integral ##\iint\limits_{\sum} f\cdot d\sigma##

    But the answer provided is ##\frac{15}{4} ## How is that? What is wrong in the above computation of answer?
  20. WMDhamnekar

    A Solve Line Integral Question | Get Math Help from Physics Forums

    I don't have any idea about how to use the hint given by the author. Author has given the answer to this question i-e F(x,y) = axy + bx + cy +d. I don't understand how did the author compute this answer. Would any member of Physics Forums enlighten me in this regard? Any math help will be...
  21. WMDhamnekar

    I How Does Angular Velocity Affect Instantaneous Acceleration in Rotating Systems?

    Homework Statement:: Find the instantaneous acceleration of a projectile fired along a line of longitude (with angular velocity of ##\gamma##constant relative to the sphere) if the sphere is rotating with angular velocity ##\omega##. Relevant Equations:: None Find the instantaneous...
  22. WMDhamnekar

    Angular Velocity in the Rotating systems

    Summary: Consider a body which is rotating with constant angular velocity ω about some axis passing through the origin. Assume the origin is fixed, and that we are sitting in a fixed coordinate system ##O_{xyz}## If ##\rho## is a vector of constant magnitude and constant direction in the...
  23. WMDhamnekar

    HP 50g calculator's answer is correct or author's answer is correct?

    Summary: Evaluate ##\displaystyle\iint\limits_R e^{\frac{x-y}{x+y}} dA ## where ##R {(x,y): x \geq 0, y \geq 0, x+y \leq 1}## Author has given the answer to this question as ## \frac{e^2 -1}{4e} =0.587600596824 ## But hp 50g pc emulator gave the answer after more than 11 minutes of time...
  24. Alexanddros81

    I I want to know which software was used to create vector calculus graph

    Hi. I have the Marsden an Tromba vector calculus book 6th edition. I was wondering which software was used to create the books graphs. I attach two graphs as an example. Thanks
  25. WMDhamnekar

    MHB Maxima and Minima (vector calculus)

    Hi, Hi, Author said If we look at the graph of $ f (x, y)= (x^2 +y^2)*e^{-(x^2+y^2)},$ as shown in the following Figure it looks like we might have a local maximum for (x, y) on the unit circle $ x^2 + y^2 = 1.$ But when I read this graph, I couldn't guess that the stated function have a...
  26. MichaelBack12

    Calculus Where Can I Find Online Courses for John Hubbard's Vector Calculus Textbook?

    Anyone know of an online course or set of video lectures on John Hubbard's textbook on Vector Calculus, Linear Algebra, and Differential Forms?
  27. C

    I Difficulty in understanding step in Deriving WKB approximation

    In Zettili book, it is given that ## \nabla^2 \psi \left( \vec{r} \right) + \dfrac{1}{\hbar ^2} p^2 \left( \vec{r} \right) \psi ( \vec{r} ) =0 ## where ## \hbar## is very small and ##p## is classical momentum. Now they assumed the ansatz that ## \psi ( \vec{r} ) = A ( \vec{r} ) e^{i S( \vec{r} )...
  28. V

    I Limit cycles, differential equations and Bendixson's criterion

    I have a few questions about the negative Bendixon criterion. In order to present my doubts, I organize this post as follows. First, I present the theorem and its interpretation. Second, I present a worked example and my doubts. The Bendixson criterion is a theorem that permits one to establish...
  29. runningphysics

    Solving Motion Equations with Integration

    I'm not sure where to start, when I tired using integration of the initial equation to get pos(t)=-.65t^2 i + .13t^2 j + 14ti +13tj but after separating each component, i and j, and setting j equal to zero I got 0 or -100 seconds which doesn't seem like a reasonable answer.
  30. chwala

    Calculate the area of the triangle- Vector Calculus

    This is the question, Now to my question, supposing the vectors were not given, can we let ##V=\vec {RQ}## and ##W=\vec {RP}##? i tried using this and i was not getting the required area. Thanks...
  31. U

    A question on the definition of the curl of a vector

    The curl is defined using Cartersian coordinates as \begin{equation} \nabla\times A = \begin{vmatrix} \hat{x} & \hat{y} & \hat{z} \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ A_x & A_y & A_z \end{vmatrix}. \end{equation} However, what are the...
  32. U

    Finding a vector from the curl of a vector

    Consider the following \begin{equation} \nabla\phi=\nabla\times \vec{A}. \end{equation} Is it possible to find ##\vec{A}## from the above equation and if so, how does one go about doing so? [Moderator's note: moved from a homework forum.]
  33. H

    A The ``kinematic equation'' of fluid flows

    I saw this in a textbook and I thought it is a corollary of Reynold's transport theorem. Let \mathbf{F} be a smooth vector field Consider the surface integral: \int_{S}\mathbf{F}\cdot d\mathbf{S} and now take the derivative of it, then the expression can be written as...
  34. Darsh_22

    How do I sketch a flow profile and solve for curl in vector calculus?

    Hello, Can someone explain how to sketch the flow profile in detail. Also, I solved for curl, but I'm getting a zero while the answer is the differentiation of the function f(y). Pls do help me out!
  35. jorgeluisharo

    Vector calculus — Computing this Divergence

    I really don't know how to proceed if I'm not using an specific coordinate system, Is there a way of doing this using only indices, in general form?
  36. L

    Stokes' theorem gives different results

    Given surface ##S## in ##\mathbb{R}^3##: $$ z = 5-x^2-y^2, 1<z<4 $$ For a vector field ##\mathbf{A} = (3y, -xz, yz^2)##. I'm trying to calculate the surface flux of the curl of the vector field ##\int \nabla \times \mathbf{A} \cdot d\mathbf{S}##. By Stokes's theorem, this should be equal the...
  37. C

    Vector calculus - show that the integral takes the form of (0, a, 0)

    Since the question asks for Cartesian coordinates, I wrote dV as 2pi(x^2+y^2+z^2)dxdydz and did the integral over the left hand side of the equation with x, y, z from 0 to R. My integral returned (0, 2*pi*R^5, 5/3*pi*R^6) which doesn't seem right. I also tried to compute the right-hand side of...
  38. F

    How to approach vector calculus identities?

    Majoring in electrical engineering imply studying Griffiths book on electrodynamics, so I have begun reading its first chapter, which is a review of vector calculus. A list of vector calculus identities is given, and I would like to derive each one, with one of them being ##\nabla \cdot (A...
  39. WMDhamnekar

    MHB Does the Cosine Rule Apply to Vector Addition in 3-D?

    Hi, In $\mathbb{R^3} || v-w ||^2=||v||^2 + ||w||^2 - 2||v||\cdot ||w||\cos{\theta}$ But can we say $||v+w||^2=||v||^2 +||w||^2 + 2||v|| \cdot||w|| \cos{\theta}$ where v and w are any two vectors in $\mathbb{R}^3$
  40. Hamiltonian

    B Basic doubts in vector and multi variable calculus

    If say we have a scalar function ##T(x,y,z)## (say the temperature in a room). then the rate at which T changes in a particular direction is given by the above equation) say You move in the ##Y##direction then ##T## does not change in the ##x## and ##z## directions hence ##dT = \frac{\partial...
  41. K

    Nabla operations, vector calculus problem

    Here is how my teacher solved this: I understand what the nabla operator does, ##∇\cdot\vec v## means that I am supposed to calculate ##\sum_{n=1}^3\frac {d\vec v} {dx_n}## where ##x_n## are cylindrical coordinates and ##\vec e_3 = \vec e_z##. I understand why ##∇\cdot\vec v = 0##, I would get...
  42. Bright Liu

    How do I derive this vector calculus identity?

    ##(\nabla\times\vec B) \times \vec B=\nabla \cdot (\vec B\vec B -\frac 1 2B^2\mathcal I)-(\nabla \cdot \vec B)\vec B## ##\mathcal I## is the unit tensor
  43. Kaguro

    Flux in a rotated cylindrical coordinate system

    ##\vec F= 2x^2y \hat i - y^2 \hat j + 4xz^2 \hat k ## ## \Rightarrow \vec \nabla \cdot \vec F= 4xy-2y+8xz## Let's shift to a rotated cylindrical system with axis on x axis: ##x \to h, y \to \rho cos \phi, z \to \rho sin \phi ## Then our flux, as given by the Divergence theorem is the volume...
  44. WMDhamnekar

    MHB Are $\vec{r}$ and $\frac{d^2\vec{r}}{dt^2}$ Parallel When m+n=1?

    Given $\vec{r}=t^m* \vec{A} +t^n*\vec{B}$ where $\vec{A}$ and $\vec{B}$ are constant vectors, How to show that if $\vec{r}$ and $\frac{d^2\vec{r}}{dt^2}$ are parallel vectors , then m+n=1, unless m=n? I don't have any idea to answer this question. If any member knows the answer to this...
  45. WMDhamnekar

    MHB How Can I Prove These Vector Calculus Relations?

    Hi, Let f(t) be a differentiable curve such that $f(t)\not= 0$ for all t. How to show that $\frac{d}{dt}\left(\frac{f(t)}{||f(t)||}\right)=\frac{f(t)\times(f'(t)\times f(t))}{||f(t)||^3}\tag{1}$ My attempt...
  46. Adesh

    I'm not getting the curl of vector potential equal to magnetic field

    In this image of Introduction to Electrodynamics by Griffiths . we have calculated the vector potential as ##\mathbf A = \frac{\mu_0 ~n~I}{2}s \hat{\phi}##. I tried taking its curl but didn't get ##\mathbf B = \mu_0~n~I \hat{z}##. In this thread, I have calculated it like this ...
  47. Adesh

    How to find the curl of a vector field which points in the theta direction?

    I have a vector field which is originallly written as $$ \mathbf A = \frac{\mu_0~n~I~r}{2} ~\hat \phi$$ and I translated it like this $$\mathbf A = 0 ~\hat{r},~~ \frac{\mu_0 ~n~I~r}{2} ~\hat{\phi} , ~~0 ~\hat{\theta}$$(##r## is the distance from origin, ##\phi## is azimuthal angle and ##\theta##...
Back
Top