# Questions tagged [vector-analysis]

Questions related to understanding line integrals, vector fields, surface integrals, the theorems of Gauss, Green and Stokes. Some related tags are (multivariable-calculus) and (differential-geometry).

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### Flux through cone

Consider the vector field $\mathbf{v}=(x^2y-\sqrt{z}, \sqrt{z}-xy^2,z)$ and a surface given by $\frac{z^2}{2}=x^2+y^2$ and $z\in[0,2]$. Calculate the flux of $\mathbf{v}$ through the surface. ...
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### Is it possible to make an integral by part on measurable sets?

I'm trying to show that a function $u \in W^{1,p}(\Omega)$, where $p \ge 1, n \ge 1$ and $\Omega\subset \mathbb{R}^n$ a bounded open. Where $u$ can be written as $$\sum_{j=1}^m u_j \mathbb{1}_{A_j}$$ ...
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### Line integration in pauls notes calculus III

Why did he use $[0,2]$ as the limits of integration in the 3ed curve after substituting the value of $x = 1$ in $x$, shouldn't the limits be $[0,1]$ as in the $X$ axis, so the last integration ...
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### Integral of a differential form

I'm reading a note about integral of a differential 1-form and would like to know more about its derivation. Is there an intuitive way to understand this formula? Something akin to "sum of weighted ...
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### Why does this “gradient field test” not work on the spin field $S / r^2$? (from Strang's Calculus)

In section 15.2, Strang's Calculus explains that for any gradient field $\bf{F} = Mi + Nj$, ${\partial M \over \partial y} = {\partial N \over \partial x}$. (Strang calls this "test D" for identifying ...
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### Existence and uniqueness of solenoidal decomposition $f=f^{\text{s}}+\nabla\phi$ for a vector field $f\colon\mathbb{R}^n\rightarrow\mathbb{R}^n$?

I am interested in the existence of a global solution to the Poisson equation $$\Delta\phi=g \quad \text{in} \ \mathbb{R}^n$$ such that $\phi\rightarrow 0$ when $|x|\rightarrow\infty$. My motivation ...
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### Green's Theorem to calculate a line integral

Use Green's Theorem to evaluate the line integral $\int_C y^3dx + x^3dy$ where $C$ is the ellipse $\frac{x^2}{9} + \frac{y^2}{25} = 1$, so I did this using symmetry. However, I friend switched to ...
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### Does the Divergence Theorem apply to surfaces with inward-facing normal vectors?

Conceptually I'm having some trouble with understanding the divergence theorem. In the derivations and definitions I'm finding, it always assumes outward facing normals. If I assume inward-facing ...