For questions on the vector operators: gradient, curl and divergence.

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### Computing flux integral in two ways what is my mistake?

So doing it this way is easy $\int_D \nabla F dV$ which gives me $8/3$ But doing it via $\int _{\delta D}F\cdot nds=\int _{\delta D}F\cdot n\left|r\left(t\right)\right|dt$ gives me troubles. After ...
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### Numerically compute and clear divergence of discrete vector field

I have a fluid simulation that represents velocity as a vector field in a grid of cells. The cells all have the same width and the same height, but the height is not necessarily equal to the width. I ...
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### Abundance of divergence-free vector fields in noncompact manifolds

Let $(M,g)$ be a complete noncompact Riemannian manifold of dimension $n \geq 2$. How big is the space $D_b(M)$ of (pointwise) bounded divergence-free vector fields on $M$ of noncompact support? I ...
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### Do line integrals of closed curves depend on the orientation of the curve or of the vector field?

I am trying to understand how circulation works as a line integral with the curl in green's theorem. I know that a line integral describes the relationship between a vector field and a path, i.e. how ...
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### How do I make my formula for $\nabla \times \mathbf{F}(x,y)$ correct?

Apparently, the curl of a vector field is a function that outputs the "rotationality" of the vector field at some point, as a function of that point's coordinates. I want to go from this ...
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### What's wrong with this integration over the volume of a sphere (from Gauss' Theorem)?

I'm struggling to find my mistake in the following problem: Let $S$ be a sphere of radius $R$ with center at the origin. Let $f(x,y,z) = 3z^2$. Find the integral of $f$ over the volume of the ...
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### Inverse of gradient of divergence, in two-dimensions?

Given a two-dimensional vector field $U(x,y) \in C^2(\mathbb{R}^2, \mathbb{R}^2)$, consider the expression \begin{align} \operatorname{grad} \operatorname{div} U(x,y) = V(x,y) \end{align} for some ...
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### Help understanding the use of tetrahedrally arranged vectors to compute the gradient of a function

As a long-time user of the free, open-source raytracer POV-Ray, I'm trying to understand some of the source code used to compute and perturb surface normals. The method uses the evaluation of 4 ...
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### When is valid to first evaluate a function in a point and then do the DIvergence than to first the divergence and then evaluate?

I have been having this doubt when applying the Divergence of a Electrical Potential evaluated in a point, and then doing the Divergence to get the Electric Field in that point. But is this valid? or ...
I am trying to verify the vector calculus identity in appendix B.2 in "Finite Elements Methods: A Practical Guide" (Whiteley 2017) \begin{aligned} \nabla \cdot (vp\nabla u) &= v\... 0 votes 0 answers 32 views ### How do I find orthogonal vectors using the del operator in an arbitrary coordinate system? enter image description here The image shows the full question, part (b) is what I'm interested in. I'm stuck on how to use the del operator here to find out if \vec{B}_0, \vec{E}_0 and \vec{q} ... 8 votes 3 answers 1k views ### Why does it seem the inner curls within a surface always cancels in order for greens theorem to be true Im trying to learn aerodynamics in general for my course. Every video i see to derive the concept of greens and stokes theorem shows how the inner curls within a surface area cancel to 0 and its only ... • 193 0 votes 0 answers 37 views ### Curl of normal unit vector of a smooth and closed surface? Let's say we have a curvilinear coordinate system (\rho,\theta,\zeta). Also, let's say we have a smooth and closed surface \Gamma parameterized as \Gamma: \mathbf{S}(\rho(\theta,\zeta),\theta,\... 1 vote 0 answers 62 views ### Local stability estimate for divergence free vector field For a divergence free field w\in [L^2(\mathbb{R}^d)]^d, especially for d = 3, it is well known that there exists a skew-symmetric matrix \psi \in [W^{1,2}_{loc}(\mathbb{R}^d)]^{d\times d} with \... • 71 0 votes 0 answers 82 views ### Formula for the divergence of a tensor I need to express the divergence of a tensor \nabla \cdot (a \times b) to a different form. I found here that\nabla \cdot (a \times b) = (\nabla \cdot a)b + a \cdot \nabla b which is exactly ...
I am looking for example applications of Green's theorem (in $2D$) that appeal to physicists or engineers. It's to come up with example for the divergence theorem in fluid dynamics, but finding a very ...