# 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...