MHB Rayan's question at Yahoo Answers (Green's Theorem)

AI Thread Summary
Green's Theorem is applied to calculate the work done by the force F on a particle moving counterclockwise around the closed path C defined by r = 2 cos(θ). The force F is given as F(x,y) = (e^x − 9y)i + (e^y + 4x)j. The path C corresponds to a circle with the equation (x-1)² + y² = 1, and the area D is the disk bounded by C. The work W is computed as 13 times the area of D, resulting in W = 13π. This calculation demonstrates the effective use of Green's Theorem in evaluating work done by vector fields.
Fernando Revilla
Gold Member
MHB
Messages
631
Reaction score
0
Here is the question:

Use Green's Theorem to calculate the work done by the force F on a particle that is moving counterclockwise around the closed path C.
F(x,y) = (e^x − 9y)i + (e^y + 4x)j
C: r = 2 cos(θ)

Here is a link to the question:

Use Green's Theorem to calculate the work done by the force F? - Yahoo! Answers

I have posted a link there to this topic so the OP can find my response.
 
Mathematics news on Phys.org
Hello Rayan,

Easily proved, $C:r=2\cos \theta$ is the circle $C:(x-1)^2+y^2=1$. If $D$ is the disk with boundary $C$, then by the Green's theorem, $$W=\int_C(e^x − 9y)dx + (e^y + 4x)dy=\iint_D(Q_x-P_y)dxdy=\iint_D(4+9)dxdy\\=13\iint_Ddxdy=13\mbox{Area }(D)=13\cdot \pi\cdot 1^2=\boxed{\;13\pi\;}$$
If you have further questions you can post them in the http://www.mathhelpboards.com/f10/ section.http://www.mathhelpboards.com/f10/
 
Suppose ,instead of the usual x,y coordinate system with an I basis vector along the x -axis and a corresponding j basis vector along the y-axis we instead have a different pair of basis vectors ,call them e and f along their respective axes. I have seen that this is an important subject in maths My question is what physical applications does such a model apply to? I am asking here because I have devoted quite a lot of time in the past to understanding convectors and the dual...
Fermat's Last Theorem has long been one of the most famous mathematical problems, and is now one of the most famous theorems. It simply states that the equation $$ a^n+b^n=c^n $$ has no solutions with positive integers if ##n>2.## It was named after Pierre de Fermat (1607-1665). The problem itself stems from the book Arithmetica by Diophantus of Alexandria. It gained popularity because Fermat noted in his copy "Cubum autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos, et...
Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. In Dirac’s Principles of Quantum Mechanics published in 1930 he introduced a “convenient notation” he referred to as a “delta function” which he treated as a continuum analog to the discrete Kronecker delta. The Kronecker delta is simply the indexed components of the identity operator in matrix algebra Source: https://www.physicsforums.com/insights/what-exactly-is-diracs-delta-function/ by...
Back
Top