Questions tagged [greens-theorem]
This tag is for questions about Green's theorem. Green's theorem gives the relationship between a line integral around a simple closed curve $C$ and a double integral over the plane region $D$ bounded by $C$.
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How to deduce the general case of Green's theorem from rectangular case?
The proof of Green's Theorem for rectangle is very simple.
The proof for triangle is not too bad, and the general case follows ( at least intuitively ) from the triangle case.
But I'm lazy.
So is ...
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Question about the use of Green's theorem in Do Carmo's proof of the local Gauss-Bonnet theorem.
This is Do Carmo's statement and proof of the local Gauss-Bonnet theorem in "Differential Geometry of Curves and Surfaces":
My question is: how can he assert that
$$\sum_{i=0}^k \int_{s_i}^{...
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integration by parts in Hilbert space
I want to prove that $u,v∈H^1 (R^n )$ we have : $∫_{R^n} \frac{∂u}{∂x_i} vdx=-∫_{R^n} \frac{∂v}{∂x_i} udx$
We know that $C_c^∞$ $(R^n )$ is dense in $H^1$ $(R^n)$, this means that $∃ u_n∈C_c^∞ (R^n )$...
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Integral calculated directly and with Gauss Green formula
I checked on Wolfram Alpha that the directly calculated integral was correct, when I go to use the Gauss Green formula, I get $0$ as a result, when instead, I ...
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What is this formula? Area of Jordan curve
I came across the following area formula in do Carmo's book:
I tried searching online for where this formula came from, but I couldn't find anything that matched this. Does anyone have a name for the ...
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Area of ellipse using Green’s theorem [duplicate]
Given the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$, which we want to calculate the area of. Parameterization $\mathbf{r}(t) = (a \cos t, b \sin t), \ t \in [0, 2\pi]$
My book says we can ...
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Finding area using green's theorem
A friend of mine gave me a variant of the goat problem, which is the following: If a goat is tied to a circular fence of radius $10$ feet with a rope of $20$ feet, how much land can the goat roam? I ...
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Calculate the integral with the Gauss Green formula
Calculate the following double integral (directly and with the Gauss Green formula)
$$\iint_D xy\,\mathrm{d}x\,\mathrm{dy}$$$$D={(x,y)|y\le x \le \sqrt{y+2}, 0\le y \le 2}$$
directly: $$\int^2_0\int^\...
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Do line integrals of closed curves depend on the orientation of the curve or of the vector field?
I am trying to understand how circulation works as a line integral with the curl in green's theorem. I know that a line integral describes the relationship between a vector field and a path, i.e. how ...
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Two "theorical" questions about Line Integrals
While the majority of exercises related to Line Integrals are more enfocated to calculous and the difficulty of finding a correct parametrization or so, i have found these two questions (which may be ...
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Line integral of a vector field over a triangle [closed]
I am trying to solve this line integral, but I get different answers when doing a normal line integral and using the vector form of green's theorem and am wondering why this is the case? I got 4 as my ...
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Line integral where $C$ is the boundary of the square -Green's Theorem
I am trying to solve the following problem but I face difficulties with the results. Any suggestion?
The line integral $\int_C y^2 dx - x dy$, where $C$ is the boundary of the square $[-1,1]\times[-1,...
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Geometrical meaning of calculating area using Green's theorem
Green's theorem says that:
$$
\int_C L \ dx + \int_C M \ dy = \iint_D \frac{\partial M}{\partial x} - \frac{\partial L}{\partial y} \ dx \ dy
$$
If the M and N statisfy $\frac{\partial M}{\partial x} -...
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The work done by the force $ \vec{F} $ on a particle
Question
The work done by the force ${\vec{F}}=(x^{2}-y^{2})\hat{i}+(x+y)\hat{j}$ in moving a particle along the closed path $C$ containing the curves $x+y=0,x^{2}+y^{2}=16$ and $y = x$ in the first ...
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Could I use Green's Theorem here?
I want to solve for the line integral:
$$\tag{1}\oint \alpha\nabla \phi_i\cdot \hat{\textbf{n}} ds$$
on the square boundary: $(0\le x \le 1, 0), (1,0 \le y \le 1), (1 \le x \le 0, 1),(0,1\le y \le 0)$
...
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