Questions tagged [vector-fields]

Vector-fields are vector valued functions which define a vector at each point in space. Examples of the vector field include the electric field and the velocity of a fluid.

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125 views

The strange character of operator $\nabla$

I was first introduced to the mathematical operation gradient, divergence and curl not in Mathematics but during my studies of Electromagnetism. As you all know learning Maths from a Physics teacher ...
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Why is the name of the “Field line” is “Field line”?

Faraday-inspired "Field lines" are not always straight. 【My question】 Why is the name of the "Field line" is "Field line", not "Field curve"? This may be a question of the English language, but ....
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Cross product analogue of the circulation

In fluid dynamics, the circulation around some closed contour $\mathcal{C}$ of a fluid with the velocity field $\mathbf{v}(\mathbf{x}, t)$ is defined as $$ \Gamma = \oint_\mathcal{C}\mathbf{v}\cdot\...
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Assignment of energy functions to flows is “equivariant”?

I am trying to understand the 2012 blog post What is a symplectic manifold, really? It says (with correction of a typo in the second point): If $f: M \to \mathbb{R}$ is a smooth compactly ...
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Prove a function is a first integral iff its Lie derivative vanishes [migrated]

Definition. $X$ smooth vector field over a manifold $M$, the Lie derivative of $f\in C^{\infty}(M)$ along the flow of $X$ is: $\mathcal L_X f(m) = \langle df, X \rangle (m)$. Definition. $M$ manifold,...
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Two questions about these notations/operations in the covariant derivative?

For two vector fields V and T we can take the covariant derivative: $$\nabla_V T=\nabla_{V^\mu \hat e_{\mu}}T$$ $$=V^\nu \nabla_{\hat e_\nu}T$$ What exactly are we doing when we take the vector ...
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Why can't magnetic field lines intersect with each other? [duplicate]

Why can't magnetic field lines intersect with each other? My teacher said that if they happen to intersect with each other then the compass needle will show two different directions at a time which is ...
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37 views

Erratum in Griffiths Electrodynamics (wrong reference is made) [duplicate]

I have the Fourth Edition of Introduction to Electrodynamics by David J. Griffiths, published by Pearson. I have found this erratum in Chapter 5 Magnetostatics, page 234 . We can see that "...
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Erratum in Griffith's Introduction to Electrodynamics

Applying the divergence to Eq. $47$, we obtain $$ \mathbf{\nabla} \cdot \mathbf{B} = \frac{\mu_{0}}{4\pi} \int \nabla \cdot \left( \mathbf{J} \times \ \frac{\hat{\mathbf{r}}}{r^2}\right) d\tau^{'}...
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Whats the difference between differentiating w.r.t Source Coordinates and differentiating w.r.t Field Coordinates?

While reading the bound charges section on Griffith(ED), I came upon the equation: $\vec{\nabla}^\prime\bigg({1\over r}\bigg)={\hat{r}\over r^2}$ And Griffith goes onto say that the prime ...
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Vector field surface integral in spherical coordinates [migrated]

I am trying to show the divergence theorem holds for $$\textbf{v}=r^2cos\theta \hat{r}+r^2cos\phi\hat{\theta}-r^2cos\theta sin\phi \hat{\phi}$$ over a spherical volume centred at the origin. So I ...
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What's the point of this killing vector notation?

Reading Sean Carroll's spacetime and geometry he says If $x^{\sigma_*}$ is the coordinate which ${\mu\nu}$ is independent of, let us consider the vector $\partial_{\sigma_*}$ which we label as $$...
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A question about magnetic field lines

In the simple experiment of spraying iron filings around a magnet, why doesn't the magnetic field (provided that it is strong enough) cause the iron filings to form a $3\mathrm D$ skeleton along the ...
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MTW's Gravitation, Box 11.2, etc.: Relative acceleration depends only on $\mathbf{u}$ and $\mathbf{n}$ at the fiducial point?

My question relates to MTW's Gravitation, Box 11.2 (copied below) and the discussion on page 271. This is my paraphrase of the gist of box 11.2: Consider a family $\Lambda$ of timelike geodesics ...
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Gauss Law in infinite slab

When applying Gauss law inside an infinite slab of mass density $\rho$, why can we ignore the mass outside of a symmetric Gaussian surface within the mass? E.g. a rectangular box. I understand that ...
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Total current for an arbitrary current density

Imagine a localized region $\mathcal{R}$ which contains a current density $\mathbf{j}$, which we take to be divergence-less, $\mathbf{\nabla\cdot j} = 0$. What is the total current associated with ...
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How to show Current a scalar quantity under coordinate transformation?

I know current is a scalar quantity because it doesn't follow the vector law of addition , but I want to prove it with transformation of coordinates. How to do that? From where to start?
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Do physical fields always have dimension of the Tangent Space at a point?

Say we only consider classical fields in 3 dimensions. In a 3 dimensional space you have scalar or vector fields, where scalar fields can be understood as vectors of one dimension. The other physical ...
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What is the analog between charges and mass with regards to vector fields? [duplicate]

A way to think about gravitational fields is that they spawn as a consequence of mass literally bending time and space. Is there an analog of this concerning charges and the electric field intensity?
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Higher dimensional version of Stoke's Theorem / Divergence theorem

I've learnt about Stokes' Theorem and the divergence theorem that relate integrals of functions over manifolds to integrals of related functions around the boundary of the manifolds but all in 3-...
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Finding killing vector

I am introducing myself to the topic of killing vectors and therefore, after doing some reading, I try to solve some easy problems. For simplicity, I do my first steps in 2D. First, I chose the 2D-...
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Scalar and vector transformation

When we say that a scalar field is invariant under any transformation, why can't we consider vector components as scalar fields ? Let's say we have $O(3)$ acting on a scalar and vector field in a ...
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Why don't we define potential due to a magnetic field?

We define electric potential and gravitational potential and use them quite often to solve problems and explain stuff. But I have never encountered magnetic potential, neither during my study (I am a ...
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73 views

Continuity Equation for fluid in a curved spacetime

The current of fluid is the vector $J^{\nu}$. In free-falling laboratory due to Equivalence principle holds the known Continuity Equation $\partial_{\nu}\,J^{\nu}=0$, where the ordinary 4-divergence ...
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Correct explanation for generic vectorial, radial and central field

I suppose to have a absolutely general field of the type $\vec{F}(\vec{r})\colon D\subseteq\Bbb R^3\to \Bbb R^3$ where $|\vec{r} |=\vec{OP}$. If I fix the direction of $\vec{r}$, than $\vec{F}(\vec{r}...
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1answer
37 views

Why are the vector potential components assumed zero when the respective current density components are?

My texbooks assumes that for example if $j_x=0$, then $A_x=0$, as a starting point to find the vector potential of a circuit with current density $\vec{j}$, but from the equation $\nabla^2 \vec{A} =-\...
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Killing vectors and conserved quantities in general relativity

I know that this question has been asked and answerd already (see for example here and here) and although the second answer comes pretty close to what my problem is (even touching upon my question one,...
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Hypersurface four-vector, or a familly of four 3-forms?

While reading my old personal notes on forms in relativity, I got confused about some aspects of the mathematical formalism (integration on tensors and p-forms). The energy-momentum flux across some ...
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Is the output of a line integral over a scalar field a vector?

In my physics book of "mathematical methods for physics", the author writes that line integral of a scalar function $\phi$ over a curve $C$ can be written as the following: $$\int_C\phi\,\text d{\...
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1answer
36 views

Why is this traversal drawn clockwise in The Feynman Lectures on Physics Vol II Fig 18-5?

In Vol II Chapter 3 Fig 3-10 of The Feynman Lectures on Physics we are shown a traversal of the perimeter of a square lying in the XY plane. That is, with the Z axis pointing out of the page. The ...
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2answers
703 views

Why is the divergence of the field zero in Maxwell's equations?

I read in a book called Vector Analysis by Murray R. Spiegel by Schaums Series, and I found that there is somewhere printed that the divergence of the electric field is zero. Since my teacher told ...
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51 views

How to draw quiver plot for complex-valued electric field?

I have a matrix of complex numbers for the electric field inside a medium. Since I want to draw the quiver plot of these elements, it will be completely different if I only use the absolute part. Then ...
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54 views

Force along $x$ direction on a bumpy surface

Gravitational potential energy near earth is given by$$U(y)=mgy$$ Suppose a bumpy surface is described as $y=\sin(x)$, then $U$ varies with $x$: $$U(x) = mg\sin x $$ Then the force along $x$ is given ...
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Why does $\vec{B}\cdot\frac{\partial \vec{B}}{\partial t}=\frac{1}{2}\frac{\partial}{\partial t} (B^2)$?

$$\vec{B}\cdot\frac{\partial \vec{B}}{\partial t}=\frac{1}{2}\frac{\partial}{\partial t} (B^2)$$ Griffiths states this result in his derivation of the Pontying vector, but I have absolutely no idea ...
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Determining whether a particle is in stable/unstable equilibrium

A force defined by $\vec F = (y^2\hat i + 2 x^2\hat j)$ is exerted on a particle which is initially at the origin of the coordinate system. The particle is placed at rest right at the origin. Is this ...
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69 views

Potential Minimum Confusion

Today my lecturer mentioned the notion of vector field and potential, he also said that if the vector field is a force field then there is a potential energy given by: $F(x)=-\dfrac{dU}{dx}$. (I have ...
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What is the physical significance/interpretation of a vanishing Lie Derivative?

In my lectures, an isometry of the metric is introduced as follows: A flow on a manifold $M$ is a one-parameter family of differomorphisms $\sigma_t:M \to M$. The flow is said to be an isometry if ...
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What vector field property means “is the curl of another vector field?”

I'm an undergraduate mathematics educator and I teach a lot of multivariable calculus. I posed this question on MSE over four years ago and I haven't gotten any definitive answers (despite 12 upvotes ...
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On the method described by Purcell for finding the magnetic field by measuring the force on a test particle

The following text is a method for finding the magnetic field as described in Purcell's Electricity and Magnetism (page $151$, the top part). Measure the force on the particle when its velocity is $...
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How do we know that the actual universe has no Killing vector fields?

This article states the following: The infinitude of conserved energies constructed via Noether’s theorem suffers a startling reversal as soon as Special Relativity is superseded by General ...
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Time-like null-like and space-like killing vector fields

If the Lie derivative of a metric tensor with respect to some vector field is zero then the vector is called killing vector (KV). The KV can be time-like, null-like and space-like. What is the ...
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Deriving $\nabla_\mu \nabla_\sigma \mathcal{K}^\rho=R^\rho_{\sigma\mu\nu}\mathcal{K}^\nu$

I want to derive this equation from Carroll's book. $$\nabla_\mu \nabla_\sigma \mathcal{K}^\rho=R^\rho_{\sigma\mu\nu}\mathcal{K}^\nu$$ We know that $\mathcal{K}^\nu$ is a killing vector and ...
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Vector potential, magnetic field and dipole moment due to a rotating cylinder

I've been struggling with the following problem: Consider a cylinder with height h and radius a with a homogeneous surface charge density $\sigma$ rotating about its symmetry axis with constant ...
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Conceptual Understanding of Zero Curl in Ampere's Law

I understand that Ampere's law tells us that the current density times $\mu_0$ at some location must be equal to the curl of $\mathbf{B}$ at that location. However, conceptually this is troubling me. ...
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1answer
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Torque experienced by a coplanar loop of current in a uniform magnetic field

There are a lot of posts on this already, but apparent all of them just consider some special case. I am now struggling with this more general case. Let there be a magnetic field with strength $B$. ...
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How to determine the constant of integration when solving a DE given by Ampere's law?

A cylindrical conductor with axis along the z-axis carries a current density $J\mathbf k$. From my understanding, $$ \mathbf B=\frac12 \mu_0 Jr\hat{\boldsymbol {\phi}}. $$ However, if we consider an ...
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The flux of a vector field

This is probably a basic question. I'm actually taking a class that introduces me to Maxwell's equations. I am currently trying to make sense of the Gauss's law and have some difficulty understanding ...
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48 views

How does vectorization affect $\nabla$?

The homework and exercise was to prove $\nabla \times {A}$ transform as a vector, and I've solved it thorough hard algebra. However, something occurred to my mind and I have a hard time to resolve it....
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66 views

Why is the strain rate tensor 1/2(gradV+gradV^T)

$$\ \\1/2\left( \nabla\mathbf{u} + (\nabla\mathbf{u}) ^\mathrm{T} \right) \ $$ Why is the strain rate tensor the equation above?
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Proof for Vector Identity

I am currently studying electrodynamic and came across the following vectoridentity, but I am unable to prove it: $$ \vec{f} \times ( \nabla \times \vec{f} ) -\vec{f}(\nabla\cdot\vec{f}) = \nabla \...

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