What is Vector field: Definition and 402 Discussions
In vector calculus and physics, a vector field is an assignment of a vector to each point in a subset of space. For instance, a vector field in the plane can be visualised as a collection of arrows with a given magnitude and direction, each attached to a point in the plane. Vector fields are often used to model, for example, the speed and direction of a moving fluid throughout space, or the strength and direction of some force, such as the magnetic or gravitational force, as it changes from one point to another point.
The elements of differential and integral calculus extend naturally to vector fields. When a vector field represents force, the line integral of a vector field represents the work done by a force moving along a path, and under this interpretation conservation of energy is exhibited as a special case of the fundamental theorem of calculus. Vector fields can usefully be thought of as representing the velocity of a moving flow in space, and this physical intuition leads to notions such as the divergence (which represents the rate of change of volume of a flow) and curl (which represents the rotation of a flow).
In coordinates, a vector field on a domain in n-dimensional Euclidean space can be represented as a vector-valued function that associates an n-tuple of real numbers to each point of the domain. This representation of a vector field depends on the coordinate system, and there is a well-defined transformation law in passing from one coordinate system to the other. Vector fields are often discussed on open subsets of Euclidean space, but also make sense on other subsets such as surfaces, where they associate an arrow tangent to the surface at each point (a tangent vector).
More generally, vector fields are defined on differentiable manifolds, which are spaces that look like Euclidean space on small scales, but may have more complicated structure on larger scales. In this setting, a vector field gives a tangent vector at each point of the manifold (that is, a section of the tangent bundle to the manifold). Vector fields are one kind of tensor field.
Been long since i studied this area...time to go back.
##F = x \cos xi -e^y j+xyz k##
For divergence i have,
##∇⋅F = (\cos x -x\sin x)i -e^y j +xy k##
and for curl,
##∇× F = \left(\dfrac{∂}{∂y}(xyz)-\dfrac{∂}{∂z}(-e^y)\right) i -\left(\dfrac{∂}{∂x}(xyz)-\dfrac{∂}{∂z}(x \cos...
The info at this link says the flowing:
I'll quote and highlight the confusing parts in bold:
"The photon field is a quantum field theory. It is a vector field because it includes spin-1 photons."
"The photon field of QFT is a gauge field. This is the more likely “photon field” discussed by...
Hi,
I am not sure if I have solved task b correctly
According to the task, ##\textbf{F}=f \vec{e}_{\rho}## which in Cartesian coordinates is ##\textbf{F}=f \vec{e}_{\rho}= \left(\begin{array}{c} \cos(\phi) \\ \sin(\phi) \end{array}\right)## since ##f \in \mathbb{R}_{\neq 0}## is constant...
hi, I have go through many books - they derive Dirac equation from Dirac Lagrangian, KG equation from scalar Lagrangian - but my question is how do we get Dirac or scalar Lagrangian at first place as our starting point - kindly help in this regard or refer some book - which clearly elaborate...
Hi, a doubt related to the calculation done in this old thread.
$$\left(L_{\mathbf{X}} \dfrac{\partial}{\partial x^i} \right)^j = -\dfrac{\partial X^j}{\partial x^i}$$
$$L_{\mathbf{X}} {T^a}_b = {(L_{\mathbf{X}} \mathbf{T})^a}_b + {T^{i}}_b \langle L_{\mathbf{X}} \mathbf{e}^a, \mathbf{e}_i...
I am working with some data which represents the fluid position and velocity for each point of measurement as an x, y, u, and v matrix (from particle image velocimetry). I have done things like circulation, and discretizing the line integral involved was no problem. I am stuck when trying to...
I am trying to compute the stress tensor defined as ##\vec{\Pi}=\eta(\nabla{\vec{u}}+\nabla{\vec{u}}^T)## where ##T## indicates the transpose.
The vector field ##\vec{u}## is defined as follows: ##\vec{u}(\vec{r})=(\frac{a}{r})^3(\vec{\omega} \times \vec{r})## with ##a## being a constant...
(a)
##\vec G=24xy\hat a_x+12(x^2+2)\hat a_y+18z^2\hat a_z## @ ##P(1,2-1)##
##\vec G=24(1)(2)\hat a_x+12(1^2+2)\hat a_y+18(-1)^2\hat a_z##
##\vec G=48\hat a_x+36\hat a_y+18\hat a_z##
(b)
I am not sure how to get this part started. Could someone point me in the right direction?
1. To find the solution simply integrate the e_r section by dr.
$$\nabla g = A$$
$$g = \int 3r^2sin v dr = r^3sinv + f(v)$$
Then integrate the e_v section similarly:
$$g = \int r^3cosv dv = r^3sinv + f(r)$$
From these we can see that ##g = r^3sinv + C##
But the answer is apparently that there...
I was thinking of using the chain rule with
dF/dx = 0i + (3xsin(3x) - cos(3x))j
and
dF/dy = 0i + 0j
but dF/dy is still a vector so how can it be inverted to get dy/dF ?
what are the other methods to calculate this?
Here is my attempt (Note:
## \left| \int_{C} f \left( z \right) \, dz \right| \leq \left| \int_C udx -vdy +ivdx +iudy \right|##
##= \left| \int_{C} \left( u+iv, -v +iu \right) \cdot \left(dx, dy \right) \right| ##
Here I am going to surround the above expression with another set of...
If we define Laplacian of scalar field in some curvilinear coordinates ## \Delta U## could we then just say what ##\Delta## is in that orthogonal coordinates and then act with the same operator on the vector field ## \Delta \vec{A}##?
So, we have A, the magnetic vector potential, and its divergence is the Lorenz gauge condition.
I want to solve for the two vector fields of F and G, and I'm wondering how I should begin##\nabla \cdot \mathbf{F}=-\nabla \cdot\frac{\partial}{\partial t}\mathbf{A} =-\frac{\partial}{\partial...
Hi,
I was reading this insight schwarzschild-geometry-part-1 about the transformation employed to rescale the Schwarzschild coordinate time ##t## to reflect the proper time ##T## of radially infalling objects (Gullstrand-Painleve coordinate time ##T##).
As far as I understand it, the vector...
These are the vector fields. I really have no idea how to see if there is a curl or not. I have been looking at the rotation of the vector fields. The fields d and e seem to have some rotation or circular paths, but I read online that curl is not about the rotation of the vector field itself...
Given the equation ##\frac{xy} 3##. It is a fact that the gradient vector function is always perpendicular to the contour graph of the origional function. However it is not so evident in the plot above. Any thought will be appreciated.
Hello! So I need to find the potential function of this Vector field
$$
\begin{matrix}
2xy -yz\\
x^2-xz\\
2z-xy
\end{matrix}
$$
Now first I tried to check if rotation is not ,since that is mandatory for the potentialfunction to exist.For that I used the jacobi matrix,and it was not...
Hello! I am suspossed to write (sketch) this particular vector field.
$$V2(r) = \frac{C}{\sqrt{x^2+y^2+z^2})^3} * (x,y,z) $$ Note that the x y z is suspossed to be a vector so they would be written vertically (one over the other) but I don't know how to write vectors and matrices in LaTeX,so...
I identified $$(\Phi_{SN})_{*})$$ as $$J_{(\Phi_{SN})}$$ where J is the Jacobian matrix in order to $$(\Phi_{SN})$$, also noticing that $$\frac{\partial}{\partial u} = \frac{\partial s}{\partial u}\frac{\partial}{\partial s} + \frac{\partial t}{\partial u} \frac{\partial}{\partial t} $$, I wrote...
I am looking at the following document. In section 2.3 they have the formula for the pushforward:
f*(X) := Tf o X o f-1
I am having trouble trying to reconcile this with the more familiar equation:
f*(X)(g ) = X(g o f)
Any help would be appreciated.
I am considering using a pair of point charges: positive and negative electric charge to model a magnetic dipole's magnetic field by just average the electric field vectors between the two charged particles where they overlap. Will that work?
In this case the + field will be vectors pointing...
In part c, plotting the vector field shows the vector field is symmetric in x and y in the sets {x=y}.
in {x=y}, the variables can be interchanged and the solution becomes
x = x°e^t
y = y°e^tHowever, these solutions do not work for anywhere except {x=y} and don't satisfy dx/dt = y and dy/dt =...
Untill now i have only been able to derive the equations of motion for this lagrangian when the field $$\phi$$ in the Euler-Lagrange equation is the covariant field $$A_{\nu}$$, which came out to be :
$$-M^2A^{\nu} = \partial^{\mu}\partial_{\mu}A^{\nu}$$
I have seen examples based on the...
For divergence: We learned to draw a circle at different locations and to see if gas is expanding/contracting. Whenever the y-coordinate is positive, the gas seems to be expanding, and it's contracting when negative. I find it hard to tell if the gas is expanding or contracting as I go to the...
I'm reading 'Core Principles of Special and General Relativity' by Luscombe, specifically the introductory section on problems with defining usual notion of differentiation for tensor fields. I'll quote the relevant part:
Since the equation above is a notational mess, here's my attempt to...
[Ref. 'Core Concepts in Special and General Relativity' by Luscombe]
Let ##M,M'## be manifolds and ##\psi:M\to M'## a diffeomorphism. Even if ##\psi## weren't a diffeomorphism, and instead just a smooth map, the coordinates of the pushback of ##\mathbf{t}\in T_p(M)##, would be related to the...
Hi,
I would like to ask for a clarification about the difference between parallel transport vs Lie dragging in the following scenario.
Take a vector field ##V## defined on spacetime manifold and a curve ##C## on it. The manifold is endowed with the metric connection (I'm aware of it does exist...
Hello,
A generic vector field ##\bf {F} (r)## is fully specified over a finite region of space once we know both its divergence and the curl:
$$\nabla \times \bf{F}= A$$
$$\nabla \cdot \bf{F}= B$$
where ##B## is a scalar field and ##\bf{A}## is a divergence free vector field. The divergence...
I have a vector field which is originallly written as $$ \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}$$(##r## is the distance from origin, ##\phi## is azimuthal angle and ##\theta##...
I am trying to solve the following problem from my textbook:
Formulate the vector field
$$
\mathbf{\overrightarrow{a}} = x_{3}\mathbf{\hat{e_{1}}} + 2x_{1}\mathbf{\hat{e_{2}}} + x_{2}\mathbf{\hat{e_{3}}}
$$
in spherical coordinates.My solution is the following:
For the unit vectors I use the...
What I've done so far:
From the problem we know that the curve c is a half-circle with radius 1 with its center at (x,y) = (0, 1).
We can rewrite x = r cos t and y = 1 + r sin t, where r = 1 and 0<t<pi. z stays the same, so z=z.
We can then write l(t) = [x(t), y(t), z ] and solve for dl/dt...
I'm going through the "Advanced Lectures on General Relativity" by G. Compère and got stuck with solving one set of conditions on the subject of asymptotic flatness. Let ##(M,g)## be ##4##-dimensional spacetime and ##(u,r,x^A)## be a chart such that the coordinate expression of ##g## is in Bondi...
The moving magnet and conductor problem is an intriguing early 20th century electromagnetics scenario famously cited by Einstein in his seminal 1905 special relativity paper.
In the magnet's frame, there's the vector field (v × B), the velocity of the ring conductor crossed with the B-field of...
Homework Statement:: F is not conservative because D is not simply connected
Relevant Equations:: Theory
Having a set which is not simply connected is a sufficient conditiond for a vector field to be not conservative?
let ##f : R^3 → R## the function ##f(x,y,z)=(\frac {x^3} {3} +y^2 z)##
let ##\gamma## :[0,## \pi ##] ##\rightarrow## ##R^3## the curve ##\gamma (t)##(cos t, t cos t, t + sin t) oriented in the direction of increasing t.
The work along ##\gamma## of the vector field F=##\nabla f## is:
what i...
My idea is to evaluate it using gauss theorem/divergence theorem.
so the divergence would be
## divF = (\cos (2x)2+2y+2-2z ( y+\cos (2x)+3) ) ##
is it correct?
In this way i'ma able to compute a triple integral on the volume given by the domain
## D = \left\{ (x, y, z) ∈ R^3 : x^2 + y^2 +...
is it correct if i use Gauss divergence theorem, computing the divergence of the vector filed,
that is :
div F =2z
then parametrising with cylindrical coordinates
##x=rcos\alpha##
##y=rsin\alpha##
z=t
1≤r≤2
0≤##\theta##≤2π
0≤t≤4
##\int_{0}^{2\pi} \int_{0}^{2} \int_{0}^{4} 2tr \, dt \, dr...
Given
##F (x, y, z) = (0, z, y)## and the surface ## \Sigma = (x,y,z)∈R^3 : x=2 y^2 z^2, 0≤y≤2, 0≤z≤1##
i have parametrised as follows
##\begin{cases}
x=2u^2v^2\\
y=u\\
z=v\\
\end{cases}##
now I find the normal vector in the following way
##\begin{vmatrix}
i & j & k \\
\frac {\partial x}...
I was thinking about this while solving an electrostatics problem. If we have a vector ##\vec V## such that ##\oint \vec V \cdot d\vec A = 0## for any enclosed area, does it imply ##\vec V = \vec 0##?
I'm currently watching lecture videos on QFT by David Tong. He is going over lorentz invariance and classical field theory. In his lecture notes he has,
$$(\partial_\mu\phi)(x) \rightarrow (\Lambda^{-1})^\nu_\mu(\partial_\nu \phi)(y)$$, where ##y = \Lambda^{-1}x##.
He mentions he uses active...
Since coordinate transformations should be one-to-one and therefore invertible, wouldn’t there be no restriction on pushforwarding or pullbacking whatever fields we feel like (within the context of coordinate transformations)?
I want to render the Earth’s Magnetic field in a software and simulate solar wind electron interaction with it. How do I calculate the magnetic strength and vector orientation at each point around the Earth up to thousands of km?
Is there a formula?
Or do I need to download a vector field from...
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...