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# 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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### 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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### 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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### Converting from magnetic scalar potential to magnetic vector potential

Suppose that I have a situation where I can use the magnetic scalar potential. Given the magnetic scalar potential, is there a simple way to calculate the magnetic vector potential from the magnetic ...
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### Vector Calculus Problem on Gradient Cross Product [migrated]

$Problem:$ If a vector function $V=V(x,y,z)$ is not irrotational, show that if there exists a scalar function $g=g(x,y,z)$ such that $gV$ is irrotational, then $$V\cdot (\nabla \times V )=0$$ ...
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### Gauss divergence theorem (GDT) in physics

Some of the statements for $GDT$ which I find in modern textbooks (both electromagnetism and multivariable calculus textbooks) are: (1) Calculus: Several variables Adams Let $D$ be a regular, ...
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### Problem with relative reference frames [closed]

I came across this problem a while ago and I'd like to solve it, but I'm not even sure where to start. How would I start this problem and what formulas do I need? I'm not asking for a solution, but ...
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### 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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### Example of two linearly independent, nowhere vanishing vector fields in $\mathbb{R}^{2}$
I knew that two linearly independent and nowhere-vanishing vector fields provide a basis for the tangent space at each point in $\mathbb{R}^{2}$. Is it necessary that these two vector fields commute? ...