# Questions tagged [stokes-theorem]

Stokes' theorem is a result about integration of differential forms.

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### Calculating Flux Across a Simple Closed Curve lying on an (x,z)-cylinder

I am having some difficulties with the problem above. The approach I'm using is the following: Use Stokes Theorem to instead show that $$\iint_{S} \text{curl (\vec{G})} \cdot \hat{n} \text{ }ds =0,$$...
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### Applying Stokes' theorem on a 1-form with it's coefficients in $\mathbb C^3$

$$\mathbf{n} = \frac{1}{\sqrt 2}(\mathbf u + i \mathbf v)$$ Here is a $\mathbf n$ is a complex vector ($\mathbf n \in \mathbb C^3$). $\mathbf v$ and $\mathbf u$ both are real $3$ vectors. I want to ...
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### What is the boundary of the surface area of a cone?

To verify Stokes' theorem on a cone, I need to calculate the line integral of the given vector field around the 1 dimensional boundary of the surface of the cone. However, I can't imagine what the ...
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### Stokes' Theorem general case

With the following lemma : Lemma : Let $f_{+},f_{-} : U \longmapsto \mathbb{R}$ be $C^{1}$ maps with $f_{-} \leq 0 \leq f_{+}$ with $U \subset \mathbb{R}^{n}$ open and bounded with $C^{1}$ boundary. ...
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### Proof of Stokes' theorem for differentiable manifolds

In Evan's well-known PDE book, he states the following in an appendix, without proof: Let $U\subset\mathbb{R}^n$ be an open, bounded set with $\partial U$ being $C^1$. Suppose $u\in C^1(\bar{U})$, ...
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### Flux of Curl with given function

Let $F$ from $R^3$ to $R$ defined by $F(x, y, z) = (x − yz, xz, y)$. Let $S$ be the surface obtained by rotating the graph of $x=2^z+3^z$ with $z ∈ [0, 1]$, around the $z$-axis (with normal vectors ...
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### Proving a closed form is exact using Stokes Theorem

Let $\eta = \sum_{i=1}^n a_i(\mathbf x)dx_i$ be a closed, $C^2$ $1$-form defined in a convex open set $E$ of $\mathbb{R}^n$. Prove that it is exact by following the outline: For $\mathbf p \in E$, ...
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### Applying Stokes Theorem Without Vector Field

Q: Evaluate $\oint_S 4x dx + 9y dy + 3(x^2 +y^2) dz$ where $S$ is the boundary of the surface $z=4-x^2-y^2$ where $x,y,z \ge 0$, oriented counterclockwise as viewed from above. I am a bit confused ...
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### what's wrong with this exercise using Stokes' Theorem?

The exercise asks to verify Stokes' Theorem for the field $F (x, y, z) = (y ^ 2 + z, x + z ^ 2, x^2+y)$ on the surface generated by the rotation of the curve $y = x^2$ with $1 \leq x \leq 2$ ...
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### Can we calculate a vector field from the pressure field with Navier Stokes?

I've been looking for weeks and can't seem to wrap my head around CFD or the computation of fluid and its movement in 2D with Navier Stokes. Without an initial velocity field, and given ONLY a ...
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### How to Prove a Special Case of Stokes' Theorem?

I am currently in Calculus 3, or Multivariable Calculus and need to prove this special case of Stokes' theorem. Please forgive me as I do need this simplified to the bones to understand the ...
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### How is obtained the formula $\int_\Omega \Psi \text{div} F dx =\int_{\partial \Omega }\Psi F \cdot \nu dS - \int_\Omega \nabla \Psi \cdot Fdx$ [closed]

How to show that this inequality holds? $\int_\Omega \Psi \text{div} F dx =\int_{\partial \Omega }\Psi F \cdot \nu dS - \int_\Omega \nabla \Psi \cdot Fdx$ Where $\psi$ is a scalar function and $F$ ...
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### Using Stokes' theorem to show the integral of curl F is zero over closed S

I am looking to show that $\int_S(\nabla\times$F$).d$A $=0$ for a closed surface S. This is straightforward to do by the Divergence Theorem, but I am being asked to do it by Stokes' Theorem. So, from ...
### Solution verification for $\int_{M}\omega =0 \iff \omega=d\alpha$
I want to prove that in a compact smooth orientable manifold $M$ (without boundary) of dimension $n$ $$\int_{M}\omega =0 \iff \omega=d\alpha,\quad \omega\in \Omega^n(M), \alpha\in \Omega^{n-1}(M).$$ ...
Integral definition: $$\textbf{n}\cdot(\nabla\times\textbf u) = \lim_{A\to 0}\frac 1A\int_C\textbf u\cdot\textbf t ds$$ I am only really interested in $(\nabla \cdot \textbf u )\cdot \textbf e_x$ ...