All Questions
77
questions
5
votes
1
answer
142
views
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 ...
- 179
2
votes
0
answers
87
views
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,...
0
votes
0
answers
38
views
Application of Green's Theorem on a polar closed curve C.
Question: Let $C$ be closed curve as the boundary of region $R$.
$C$ is defined as the polar coordinate inequalities $1\le r\le2, 0\le t\le \pi$.
Define the field $F(x,y)=P(x,y)i+Q(x,y)j$ where $P=x^2+...
- 135
3
votes
0
answers
79
views
Help for evaluating the line integrals with Green's Theorem
Earlier, when I scrolled the Instagram posts I found a mathematical problem uploaded by The Vegan Math Guy like the following because this problem looks interesting to me to be solved. The ...
- 155
0
votes
0
answers
24
views
inferring sign of a double integral over a general region D from double integrals of all boxes that lie inside D and containing D
Suppose a double integral
$$
I_{D}:=\iint\limits_D f(x,y) dx dy
$$
over a specific type of region $D=\{(x,y) : x_{\min}\leq x \leq x_{\max}, h_1(x) \leq y \leq h_2(x)\}$. Note that $h_1, h_2$ are ...
- 71
3
votes
1
answer
351
views
Calculating a line integral using Green's theorem in a region with a singularity
Problem:
Calculate the line integral
$$
\int_{A}\frac{y\,dx-(x+1)\,dy}{x^2+y^2+2x+1}
$$
where $A$ is the line $|x|+|y|=4$, travelling clockwise and making one rotation.
Answer: $-2\pi$
Solution:
The ...
- 89
0
votes
0
answers
33
views
Does the starting/end point matter in line intergral of vector fields over a closed curve
Suppose I have a vector field $F = (F_1, F_2):\mathbb{R}^2 \to \mathbb{R}^2$ and $C$ a positively oriented smooth simple closed curve. Then I know the definition of
$$
\int_C F_1 dx + F_2 dy = \int_{...
- 2,913
1
vote
0
answers
47
views
Integral applying Green's Theorem
I have a vector field $\vec{F} = (-2y, x)$ and the curve $\{(x, y) : x^2+y^2 \leq 1; y > |x|\}$. Now I want to calculate the work done by the vector field along the curve that goes positively.
So, ...
- 653
0
votes
1
answer
75
views
Line integral on shifted circle directly and using Green's theorem
I am trying to compute the line integral $\int_C \omega$, where $\omega=-y\sqrt{x^2+y^2}dx+x\sqrt{x^2+y^2}dy$ and $C$ is the circle $x^2+y^2=2x$ directly and using Green's Theorem.
I have managed to ...
- 4,098
2
votes
2
answers
123
views
Green's theorem and translated angular form on the path $\gamma:[0,3\pi/2]\to\Bbb R^2,\gamma(t)=(t,\pi\cos t)$-strange result
I would like to compute the following integral $$\int_\gamma-\frac{y}{(x-\pi)^2+y^2}dx+\frac{x-\pi}{(x-\pi)^2+y^2}dy,$$ where $\gamma:\left[0,\frac{3\pi}2\right]\to\Bbb R^2,\gamma(t)=(t,\pi\cos t).$
...
- 4,670
1
vote
1
answer
50
views
Computing areas using Green's theorem
I want to compute the area of the surface $B$ with boundary parametrised by
$$
\gamma(t)=\left(\begin{array}{c}
\sin t \\
4 \cos ^{2} t+\cos t
\end{array}\right), \quad t \in[0,2 \pi]
$$
...
- 575
0
votes
0
answers
73
views
Using Green's theorem.
I'm asked to find the following integral by using Green's theorem:
$$
I=\iint_D \frac{2ay-y^2-x^2}{y}dxdy
$$
where $D$ is the region enclosed by the curve $x^2+y^2=2ay$ and $y\geq a$.
The only thing ...
- 11
0
votes
1
answer
62
views
Calculate $\int_{\rm C}(e^{x^2}- y{x}^{2}){\rm d}x+ \cos({y}^{2}){\rm d}y$
I am trying to calculate the curve integral
$$\int_{\rm C}(e^{x^2}- y{x}^{2}){\rm d}x+ \cos({y}^{2}){\rm d}y$$
where ${\rm C}$ is the unit circle passed through one lap counter clockwise. I am using ...
- 113
2
votes
1
answer
57
views
compute line integral using Greens theorem
The question is:
$$\int_\gamma \frac{(x^2+y^2-2)\ dx+(4y-x^2-y^2-2)\ dy}{x^2+y^2-2x-2y+2}$$
$$\gamma:y=2\sin\frac{\pi x}{2} \quad \text{from}\ (2,0)\ \text{to}\ (0,0)$$
Here how i have tried to solve ...
- 388
0
votes
1
answer
67
views
Why aren't the integrand $(-y\,dx+x\,dy)$ path independent?
Use Green's theorem to prove: if the vertices of the polygon in counterclockwise order are $(x_i,y_i)$, $i=1(1)n$ then the area of the polygon is
$$A=\frac 12 \sum_{i=1}^n(x_iy_{i+1}-x_{i+1}y_i).$$ $...
- 9,098