What is Green's theorem: Definition and 134 Discussions
In vector calculus, Green's theorem relates a line integral around a simple closed curve C to a double integral over the plane region D bounded by C. It is the two-dimensional special case of Stokes' theorem.
My lines are as follows;
##y=\sqrt x## and ##y=x^2## intersect at ##(0,0## and ##(1,1)##.
Along ##y=\sqrt x##, from ##(0,0)## to ##(1,1)## the line integral equals,
$$\int_0^1 [3x^2-8x] dx + \dfrac{4\sqrt x-6x\sqrt x}{2\sqrt x} dx $$
$$=\int_0^1[3x^2-8x+2-3x]dx=\int_0^1[3x^2-11x+2]dx =...
Author's answer:
Recognizing that this integral is simply a vector line integral of the vector field ##F=(x^2−y^2)i+(x^2+y^2)j## over the closed, simple curve c given by the edge of the unit square, one sees that ##(x^2−y^2)dx+(x^2+y^2)dy=F\cdot ds##
is just a differentiable 1-form. The...
Greetings!
My question is: is it possible to use the green theorem to compute the circulation while in presence of a scalar function ? I know how to solve by parametrising each part but just in case we can go faster? thank you!
Hi everyone, I was wondering if it was possible to calculate a double integral by converting it to a line integral, using the greens theorem, and if so is it possible to get a non zero answer. if we were working on a rectangular region
Homework Statement
C is the directed curve forming the triangle (0, 0, 0) to (0, 1, 1) to (1, 1, 1) to (0, 0, 0).
Let F=(x,xy,xz) Find ∫F·ds.
Homework EquationsThe Attempt at a Solution
My intial instinct was to check if it was conservative. Upon calculating:
∇xF=(0,-z,y)
I concluded that...
Homework Statement
Evaluate the line integral of (sin x + y) dx + (3x + y) dy on the path connecting A(0, 0) to B(2, 2) to C(2, 4) to D(0, 6). A sketch will be useful.
Homework Equations
Sketching the points, I have created a parallelogram shape. I also know that green's theorem formula, given...
Homework Statement
Use Green's Theorem to find the area of the region between the x-axis and the curve parameterized by r(t)=<t-sin(t), 1-cos(t)>, 0 <= t <= 2pi
Attached is a figure pertaining to the question
Homework Equations
[/B]
The Attempt at a Solution
Using the parameterized...
Homework Statement
You have inherited a tract of land whose boundary is described as follows. ”From the oak tree in front of the house, go 1000 yards NE, then 1200 yards NW, then 800 yards S, and then back to the oak tree.
Homework Equations
Line integral of Pdx + Qdy = Double integral of...
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I want to compute the integral $\oint_C \cos \left (x^{2017}\right )dx+\left (\frac{x^2}{2}+\sin y^{2018}\right )dy$, where $C$ is the boundary of the bounded field that is defined by the curves $y=2-x^2$ and $y=x$, with positive orientation.
We have to apply Green's Theorem, or not...
Please recommend two textbooks about Poisson equation, Green's function and Green's theorem for a theoretical physics student. One is easy to read so that I can have an overall understanding of the topics, another is mathematically rigorous and has a deep and modern exploration of these topics...
So here's the question:
You are given that F is a conservative vector field, except for singularities
at the points (0,1), (2,0), (3,0), and (0,4). You are given the following information
about line integrals around the following closed paths:
1) Around the curve C1 given by x^2 + y^2 = 2...
Hi. I was trying to translate the divergence theorem and the Green's theorem to tensor notation that we use in Relativity. For the divergence theorem, it was easy (please tell me if I'm wrong in the below derivation). I'm using the standard electromagnetic tensor ##F_{\mu \nu}## in place of the...
Homework Statement
Homework Equations
Green's theorem
The Attempt at a Solution
DO I first parametrize? For 1st part, I have 3 parametrizations, which I can then find the normal vector, and use in the integrals?
I have tried to apply greens theorem with P(x,y)=-y and Q(x,y)=x, and gotten ∫ F • ds = 2*Area(D), where F(x,y)=(P,Q) ===> Area(D) = 1/2 ∫ F • ds = 1/2 ∫ (-y,x) • n ds . This is pretty much the most common approach to an area of region problem. But here they ask you to prove this bizarre...
Homework Statement
I have a linear integral (e^xsiny-2)dx + (e^xcosy+x^2)dy
y≥0
2x=x^2+y^2
I used Green's theorem and got:
∬ (e^xcosy+2x) - (e^xcosy) dy dx
x bounds: from 0 to 2
y bounds: from 0 to sqrt(2x-x^2)
After solving all that stuff I get to:
∫ (2x) (sqrt(2x-x^2)) dx
x bounds: 0 to...
(Sorry for my bad English.) I was reading about the Green's theorem and I notice that the book only shows for the case where the function is a vector function. When learning about line integrals, I saw that we can do line integrals using "ordinary" functions. For example, the line integral of...
Homework Statement
An infinite plane in z=0 is held in potential 0, except a square sheet -2a<x,y<2a which is held in potential V.
Above it in z=d there is a grounded plane. Find:
a) the potential in 0<z<d?
b) the total induced charge on the z=0 plane.
Homework Equations
Green's function for a...
Homework Statement
An infinite plane in z=0 is held in potential 0, except a square sheet -2a<x,y<2a which is held in potential V.
Above it in z=d there is a grounded plane. Find:
a) the potential in 0<z<d?
b) the total induced charge on the z=0 plane.
Homework Equations
Green's function for a...
Hey Guys!
I was working on an integration problem, and I "simplified" the integral to the following:
$$\int \limits_0^{2\pi} (7.625+.275 \cos(4x))^{1.5} \cdot (A \cos(Nx) + B \sin(Nx)) \cdot (Z-v \cos(x)) dx$$
This integral may seem impossible (I have almost lost all hope on doing this...
As required by the Green's identity, the integrated function has to be smooth and continuous in the integration region Ω.
How about if the function is just discontinuous at the boundary? Actually, this function is an electric field. So its tangential component is naturally continuous, but the...
Homework Statement
Find the circulation (line integral) of y2dx+x2dy for the boundary of a triangular region contained within x+y=1, x=0, and y=0.
Homework Equations
Green's theorem
The Attempt at a Solution
I think I actually already got the solution; I used the Green's theorem to get the...
As required by the Green's identity, the integrated function has to be smooth and continuous in the integration region Ω.
How about if the function is just discontinuous at the boundary?
For example, I intend to make a volume integration of a product of electric fields, the field function is...
Homework Statement
Use Green''s Theorem in the plane to check:
\oint_C (xy+y^2) \> dx + x^2 \> dy
Where C is the closed curveof the region bound between the curve of y=x^2 and the line y=x
Homework Equations
\oint_C u \> dx + v \> dy = \int \int_A (\partial_x v - \partial_y u) \> dx \> dy...
Homework Statement
A closed curve C is described by the following equations in a Cartesian coordinate system:
where the parameter t runs monotonically from 0 to 2π, thus defining the direction of C. Calculate the area vector of the planar region enclosed by C, using the formula:
2. The...
Homework Statement
Firstly, I was seeking any clarification on whether I've made any mistakes. Secondly, further insight into Green's Theorem, if my working is all good.
I've been reading the mathinsight.org on Subtleties about curl: http://mathinsight.org/curl_subtleties
Regarding the...
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Prove using Green's theorem that the boundary value problem $$\frac{\partial}{\partial{x}}\left ( (1+x^2)\frac{\partial{u}}{\partial{x}}\right )+\frac{\partial}{\partial{y}}\left ( (1+x^2+y^2)\frac{\partial{u}}{\partial{y}}\right ) -(1+x^2+y^4)u=f(x,y), x^2+y^2<1 \\ u(x, y)=g(x,y)...
So if we have a non-simply-connected region, like this one
to apply Green's Theorem we must orient the C curves so that the region D is always on the left of the curve as the curve is traversed.
Why is this? I have seen some proofs of Green's Theorem for simply connected regions, and I...
Homework Statement
1) How do I know when to use Green's Theorem, the Fundamental Theorem for Line Integrals or the regular method of using parametrization?
2) Assuming that the three methods above are all used to solve line integrals, why do the Fundamental Theorem and Green give different...
Homework Statement
Use Green's Theorem to evaluate the line integral along the given positively oriented curve.
1) Is the statement above the same as finding the area enclosed?
2) ##\int_C \cos ydx + x^2\sin ydy ##, C is the rectangle with vertices (0,0) (5,0) (5,2) and (0,2).
3) ##\int_C y^4...
Homework Statement
Find the area of the right leaf of the Lemniscate of Gerono (the ∞ sign, see figure below) parametrized by
r(t)= <sin(t), sin(t)cos(t)>
from 0=<t=<pi
Picture is uploaded.
Homework Equations
Green's theorem: integral of fdx+gdy = double integral (over the region) of (gx-fy)...
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If we have the initial and boundary value problem $$u_{tt}(x,t)-c^2u_{xx}(x,t)=0, x>0, t>0 \\ u(0,t)=0 \\ u(x,0)=f(x), x\geq 0 \\ u_t(x,0)=g(x)$$ and we want to apply Green's theorem do we have to expand the problem to $x \in \mathbb{R}$ ?? (Wondering)
Homework Statement
Homework Equations
n/a
The Attempt at a Solution
I don't understand how the book went from calculating Green's theorem on ##\int _c Pdx + Qdy + \int _{-c'} Pdx + Qdy = ## (1 in the attached picture) to getting (labeled 2) ##\int _c Pdx + Qdy = \int _{c'} Pdx + Qdy ##...
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I want to find the solution of the following initial value problem:
$$u_{tt}(x, t)-u_{xt}(x, t)=f(x, t), x \in \mathbb{R}, t>0 \\ u(x, 0)=0, x \in \mathbb{R} \\ u_t(x, 0)=0, x \in \mathbb{R}$$
using Green's theorem but I got stuck... I found the following example in my notes...
Homework Statement
Use, using the result that for a simple closed curve C in the plane the area enclosed is:
A = (1/2)∫(x dy - y dx) to find the area inside the curve x^(2/3) + y^(2/3) = 4
Homework Equations
Green's Theorem:
∫P dx + Q dy = ∫∫ dQ/dx - dP/dy
The Attempt at a Solution
I...
Homework Statement
Let C be the closed, piecewise smooth curve formed by traveling in straight lines between the points (-2,1), (-2,-3), (1,-1), (1,5) and back to (-2,1), in that order. Use Green's Theorem to evaluate the integral ∫(2xy)dx+(xy2)dy.
Homework Equations
Green's Theorem:
∫Pdx+Qdy...
Hi PF!
So let's say I have some vector field, call it ##\vec{F}## and let ##\vec{F}## be 2-Dimensional and suppose I wanted to compute ##\iint_D \nabla \cdot \vec{F} dD##. Using green's theorem we could write ##\iint_D \nabla \cdot \vec{F} dD = - \int_{\partial D} \vec{F} \cdot \hat n dS##...
Hello! (Wave)
I have a question..
There are three formulas of the Green Theorem:
$$\oint_S (Mdx+Ndy)=\iint_R \left( \frac{\partial{N}}{\partial{x}}-\frac{\partial{M}}{\partial{y}} \right) dxdy$$
$$\oint \overrightarrow{F} \cdot d \overrightarrow{R}=\iint_R \nabla \times \overrightarrow{F}...
Homework Statement
Hello,
I know this might be trivial but,
can you please tell me what I am missing?
Here is my problem :
Homework Equations
Given F = (yi+x3j).dr
Q = x3
P = y
=> ∂Q/∂x = 3x2 and ∂P/dy = 1
∫c (3x2 -1).dr = ∫2Pi0∫10 (3r2cosθ2-1) rdrdθ
∫2Pi0∫10 (3r3cosθ2-r) drdθ
Is it true...
Homework Statement
Use Green’s Theorem to find the area of the region between the x – axis and one arch of the cycloid
parameterized by p(t) = < t-2sin(t),2-2cos(t)> for 0≤t≤2∏
p
Homework Equations
The Attempt at a Solution
My problem here is that I get different answers depending on if I...
Homework Statement
∫Fdr
Over C where C is the cirlce (x-3)^2+ (y+4)^2=4
F=<y-cosy, xsiny>
Homework Equations
The Attempt at a Solution
So I applied Green's theorem and converted to polar and ended up with -4π, it should be positive.
The orientation confused me since day one...
Homework Statement
Problem in attachment.
Homework Equations
The Attempt at a Solution
Unfortunately I was unable to attend my only class where my proffessor taught this method of solving area. Plus my prof and classmates won't help me.
Does anybody know how to solve area...
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I have to verify the Green's Theorem $\oint_C{ \overrightarrow{F} \cdot \hat{n}}ds=\iint_R{\nabla \cdot \overrightarrow{F}}dA$.
The following are given:
$$\overrightarrow{F}=-y \hat{\imath}+x \hat{\jmath}$$
$$C: r=a \cos{t} \hat{ \imath}+a \sin{t} \hat{\jmath}, 0 \leq t \leq 2 \pi$$...
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$\overrightarrow{F}=M \hat{i}+ N \hat{j}+ P \hat{k}$
To prove the Stokes' Theorem we apply Green's Theroem at $ABE$, $BCE$, $CDE$.
$(\oint_{ABE}+\oint_{BCE}+\oint_{CDE}){ \overrightarrow{F}}d \overrightarrow{R}=\iint_{ABCDE}{ \nabla \times \overrightarrow{F} \cdot \hat{n}}d \sigma$...
In wiki there is the follows formula:
https://en.wikipedia.org/wiki/Green%27s_theorem#Area_Calculation
But, I don't understand why M = x and L = -y. I don't found this step in anywhere.
Homework Statement
Verify Green's Theorem in the plane for the \oint [(x^{2} - xy^{2})dx + (y^{3} + 2xy)dy] where C is a rectangle with vertices at (-1,-2), (1,-2), (1,1) and (-1,1).
The Attempt at a Solution
This means you have to use green's theorem to convert it into a double...
I’m having a little trouble understanding why Green’s Theorem is defined as;
∮_C P dx+Q dy = ∬_D [(δQ/δx)-(δP/δy)] dA
Instead of;
∮_C P dx+Q dy = ∬_D [(δQ/δx)+(δP/δy)] dA
When proving the theorem, in the first step you simply reverse the bounds of the second integral to get the...
Homework Statement
Homework Equations
The Attempt at a Solution
$$Q=x\quad P=y^2-2y\\\oint_C{Pdx+Qdy}\\=\int_{C1}(y^2-2y)dx+xdy+\int_{C_2}(y^2-2y)dx+xdy\\=\int_{-\pi/2}^{\pi/2}(((sint+1)^2-2(sint+1))(-sint))dt+cost(cost)dt\\=\int_{-\pi/2}^{\pi/2}(2sin^2t+4sint+2)\\=3\pi$$
Correct answer is...
Homework Statement .
Calculate by a line integral the following double integral:
##\iint\limits_D (y^{2}e^{xy}-x^{2}e^{xy})dxdy##, D being the unit disk.
The attempt at a solution.
Well, if we consider C to be the curve that encloses the region D (C is the unit circle), then C is a...
Problem:
##\int_R (x-y)dx \ dy=-2/3 ## for ##R=\{(x,y):x^2+y^2 \geq 1; y \geq 0\}## by
a.) Direction integration,
b.) Green's theorem.
Attempt at a Solution:
I'm a little confused with part a. Wouldn't the region R be defined by all the points above the y-axis that lie on, in addition to...