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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I have tried to solve this question about stokes theorem but i cannot do it.

Using Stoke's theorem, evaluate the integral $$ \int_C y\, dx + 2xy\, dy + 5\,dz $$ where $C$ is the projection curve of the cone $z=4-\sqrt{x^2 +y^2}$, $x^2+y^2 \leq 1$ with outward orientation on $...
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Stokes's Theorem: Show that the integral is independent from the surface

Problem: Show that the value of $\Omega=\iint_{\Sigma}\dfrac{(a-x)dydz+(b-y)dzdx+(c-z)dxdy}{[(a-x)^2+(b-y)^2+(c-z)^2]^{3/2}}$ is independent of the choice of the surface $\Sigma$, provided its ...
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Computing a line integral

Let $C$ be the curve which obtains from the intersection of the plane $z=x$ and the cylinder $x^{2}+y^{2}=1$, oriented counterclockwise. If $F$ is a vector field $F=(x,z,2y) \in \mathbb{R}^{3}$, ...
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application of Stokes Theorem

could you help me with the following please: compute the integral $\iint_{S} (\nabla \times F)dS$, where S is the portion of the surface of a sphere defined by, $x^{2} +y^{2}+z^{2}=1$, and $x+y+z \geq ...
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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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Stokes' Theorem to solve closed line integral in space

Find the value of line integral $ \oint_{C}^{} \textbf{F} \cdot d\textbf{r} $, where $ \textbf{F} = (y-x) {\boldsymbol{\hat{\textbf{i}}}} + (z-x) {\boldsymbol{\hat{\textbf{j}}}} + (x-y) {\boldsymbol{\...
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Simplicial Stokes theorem

I have been intrested lately in a simplicial version of Stokes theorem, which intuitively I think should be true in simplicial settings. I think if $X$ is a simplicial complex and $X(k)$ are its $k$-...
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Prove that $F$ mapping $V(\Omega\subset V)$ to $\partial \Omega$ can not be the identity mapping on $\partial\Omega$

Let $\Omega\subset\mathbb R^n$ be compact and let $\partial\Omega$ be smooth. Let $V\subset\mathbb R^n$ be open and $\Omega\subset V$.Suppose $int\Omega\neq\emptyset$. If $F\in C^1(V,\partial\Omega)$,...
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Stokes's Theorem for the Cone

Consider the vector field $$F = \biggl \langle \sin x-\frac{y^3}3, \cos y+\frac{x^3}3, xyz \biggr \rangle.$$ Let $S$ be the surface given by the cone $$z^2 = x^2 + y^2 \text{ for } 0 \leq z \leq1.$$ ...
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Where to find reference for the energy method $\lambda u- \Delta u=f(x) \hspace{1cm} x \in \Omega, \hspace{1cm}$

Why do I need to multiply by the function w in the energy method to guaranty at most one solution? This is the example $\lambda u- \Delta u=f(x) \hspace{1cm} x \in \Omega, \hspace{1cm} u=0 \hspace{0....
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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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Why this does not have a dot product on the right: $\int_\Omega \nabla \phi dx =\int_{\partial \Omega}\phi \nu ds$

My main concern is because, on the right side of the expression, there is a vector $\nu$, but there is not any dot product. $$\int_\Omega \nabla \phi dx =\int_{\partial \Omega}\phi \nu ds$$ Where $\...
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Let $F=\langle xy^2, 3z-xy^2, 4y-x^2y\rangle$ Find the maximum value of the line integral of $F$ over a simply closed curve C in the plane $x+y+z=1$

Let $F=(xy^2, 3z-xy^2, 4y-x^2y)$ . Find the maximum value of the line integral of F over a simply closed curve $C$ in the plane $x+y+z=1$. What is the curve that maximises it? I am a bit confused on ...
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Using Stokes' Theorem to find the flux on F

I have been working on a Stokes' Theorem problem. (Thank you @Prasiortle for your help in [this problem.][1]) This is the question and my work so far. The reason why I'm feeling hesitant is that ...
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Using Stokes' Theorem to solve a problem

I am working on this problem: Use Stoke's Theorem to evaluate $\int_CF\bullet dr$. $C$ is the boundary of the portion of the paraboloid $x=y^2+z^2$ with $x\geq 4$, n to the back, $F= \langle yz,y-...
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Stokes theorem to find integral of differential form

Let $$\alpha=2w dx\land dy\land dz + z dx\land dy\land dw + ydx\land dz \land dw + xdy\land dz\land dw.$$ Use Stokes theorem to calculate $\int_{\partial S}\alpha$ where $S$ is a domain bounded by $x=...
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An Application of Stokes' theorem

Let $\Sigma$ be a curve in $\mathbb{R}^{3}$ with normal vector $\overrightarrow{n}$ and $\overrightarrow{a}$ be a constant vector. Prove that $$\int_{\partial\Sigma}\vec{a}\times\vec{p}\cdot\text{d}\...
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Does any oriented atlas of a manifold have maps diffeomorphic to either $\mathbb{R}^n$ or the upper half plane in $\mathbb{R}^n$?

I came across this claim: let $M$ be an oriented manifold of $\dim M=n$ and $\mathcal{A}$ an atlas for $M$. Then any $U \in \mathcal{A}$ is diffeomorphic to either $\mathbb{R}^n$ or $\mathbb{H}^n:= \...
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Application of Stokes Teorem

I need help with the following demo: Show that for a steerable surface $ \Sigma $ with $\partial \Sigma=C$ an oriented curve has to $\int_{C} (f \nabla g+g \nabla f ) d \vec{r}=0$ for any functions ...
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How to apply Stokes's Theorem properly to this cylinder

The equation for an open cylinder is $x^2+y^2=1$; where $0\leq z \leq 3$. The vector field is $\mathbf{F} = yx^2\mathbf{i}+z^3\mathbf{j}+y+z\mathbf{k}$. How do you apply Stokes's theorem to find ...
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Is this a particular case of the Stoke's Theorem? How to prove the equality?

I'm doing my Calculus III homework and I'm stuck in a question. It seems to be a particular case of the Stoke's Theorem but I'm not sure. The problem is: Be $B$ a triangle with vertex $(0,0)$, $(1,0)$...
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Verify Stokes' Theorem for Hemisphere

I am trying to answer the following question, but am having difficulty getting the same result for each side of Stokes' theorem. Question: Verify Stokes' Theorem for the hemisphere $D: x^2 + y^2 + z^...
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Integral of a closed form over an compact manifold

If $w$ is closed ($dw=0$), for a compact manifold $M$, why the integral $\int_M w$ is topological invariant instead of 0? Because from Storke theorem, I would expect that $\int_M w=\int_{Volume~...
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Determine the circulation of vector field along a boundary - Stokes' theorem

I have the following problem: Given the vector field $V(x,y,z)=\bigg(x-4y,y^2,2x-3z \bigg)$ and a surface with the parametric equation, $r(u,v)=\bigg(\sin(u)\cos(v),\sin(u)\sin(v),\cos(u) \bigg),u\in[...
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Verify Stokes' Theorem for a hemisphere where the radius is not 1

I am trying to verify Stokes' Theorem for a hemisphere with radius 3. I have only worked and found examples where the unit sphere has been used and I'm not sure how to factor in the value of the ...
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Boundaries for Stokes' Theorem

I am confused about the boundary required for Stokes' theorem to hold. Most of the time, examples I have encountered in textbooks and school courses show examples of the theorem holding for ...
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calculating the circulation of a vector field using the Stoke's theorem

$\vec{A} = y\vec{i} + y^2\vec{j}+(z-x)\vec{k}$ - the equation of a vector field $\left\{\begin{matrix}x=cos(t) \\ y=cos(t) \\ z=sin(t) \end{matrix}\right. $ this is the contour equation I was able ...
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Divergence theorem for non-compact manifolds

I wonder whether there is a generalization of the divergence theorem or more generally of Stokes' theorem to non-compact domains or manifolds, much like the improper Riemann integrals. Consider the ...
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Evaluate $\iiint_V \sqrt{x^2+y^2+z^2}\, dV$ where $V: x^2 + y^2 + z^2 \leq 2z$

Evaluate $$I=\iiint_V \sqrt{x^2+y^2+z^2}\, dV\,,$$ where $V: x^2 + y^2 + z^2 \leq 2z$. I tried using the cylindrical coordinates to arrive at: $$I = \int\int_R\int_{z=1-\sqrt{1-r^2}}^{1+\sqrt{1-r^2}}...
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Evaluate the line integral $\int_C \vec{F}.\vec{dr}$ $\vec{F} =\langle2y, xz, x+y\rangle$

Evaluate the line integral $\int_C \vec{F}.\vec{dr}$ where $\vec{F} = \langle2y, xz, x+y\rangle$ and $C$ is the curve of intersection of the plane $z = y+2$ and $x^2 + y^2 = 1$ with orientation as ...
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Boundary of a surface z=\sin(x)\cos(y)

I am doing a question on Stoke's theorem, and one of the requirements is to find the boundary of the surface $$z=sin(x)cos(y), 0{\leq}x{\leq}\pi, 0{\leq}y{\leq}\frac{\pi}{2}$$ So far, I think that ...
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Stokes Theorem: manifolds vs. chains

So, reading both Baby Rudin (Principles of Mathematical Analysis) and Munkres (Analysis on Manifolds), I start to wonder the difference between this two approaches. In Rudin's, he defines the Stokes ...
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Verifying Stoke's Theorem using path integration along a parametrized ellipse

Verify the Stoke's Theorem by showing that the line integral $$ \int\limits_{C} -y^3 dx + x^3 dy - zdz = \frac{3\pi}{2} \tag{1},$$ through direct computation, where $C$ is the intersection of the ...
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calculate line integral directly and via stokes theorem

I've calculated line integral $\int_cydx+zdy+xdx,$ where $c$ is intersection of $(x^2+y^2)^2=y^2-x^2\ (y\geq 0)$ and $z=\sqrt{x^2+y^2}$ positively oriented when looking from the point $(0,1,0)$, ...
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How does this equation $x^2+y^2+xy=1/2$ represent an ellipse?

I want to find the intersection of the sphere $x^2+y^2+z^2 = 1$ and the plane $x+y+z=0$. $z=-(x+y)$ that gives $x^2+y^2+xy= \frac 12$ How do I represent this in the standard form of ellipse? Any ...
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Help proving Stokes' theorem for manifolds without a boundary

The sources I have looked at seem to take this as being sufficiently obvious that intermediate steps don't need to be given, but I am struggling to understand why one of the integral components in ...
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Stokes theorem on Cone

We wish to verify Stokes theorem for a Cone: $z=\sqrt{x^2+y^2}$, $z\leq 3$ for ${\bf F}=\left [y , -x ,2z \ \right ]$ Curl: $\nabla \times {\bf F} =[0,0,-2] $. $\int\int_S(\nabla \times {\bf 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 ...
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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 $n$ $$\int_{M}\omega =0 \iff \omega=d\alpha,\quad \omega\in \Omega^n(M), \alpha\in \Omega^{n-1}(M).$$ ...
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Obtain the recipe for the curl of a vector u(x) in Cartesian coordinates from the integral definition given below.

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 $(\nabla \cdot \textbf u )\cdot \textbf e_x$ ...

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