Green's Theorem and a triangle

In summary, Green's Theorem is a mathematical theorem that relates a line integral around a simple closed curve to a double integral over the region enclosed by the curve. It is named after mathematician George Green and is often used in vector calculus and physics. In a triangle, Green's Theorem is used to calculate the area of the triangle by integrating over the boundary of the triangle. It is also related to the divergence theorem, which is a generalization of Green's Theorem in three dimensions. Green's Theorem has many practical applications in physics, fluid dynamics, and engineering, and is often preferred over other methods for calculating line integrals due to its simplicity and versatility.
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Homework Statement


Use Green's Theorem to evaluate ∫F*dr.
(Check the orientation of the curve before you applying the theorem.)

F(x,y)=<y2cos(x), x2+2ysin(x)>

C is the triangle from (0,0) to (2,6) to (2,0) to (0,0)

*=dot product

Homework Equations



Green's Theorem

The Attempt at a Solution



-∫∫2x+2ycos(x)-2ycos(x) dydx= -∫∫[tex]_{D}[/tex] 2x dydx

D={0[tex]\leq[/tex]y[tex]\leq[/tex]3x, and 0[tex]\leq[/tex]x[tex]\leq[/tex]2)

I put the negative out front because the problem goes from point to point in a clockwise direction. Is this set up correct?
Thank you for your time.
 
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  • #2
Looks ok to me.
 

1. What is Green's Theorem?

Green's Theorem is a mathematical theorem that relates a line integral around a simple closed curve to a double integral over the region enclosed by the curve. It is named after mathematician George Green and is often used in vector calculus and physics.

2. How is Green's Theorem applied to a triangle?

In a triangle, Green's Theorem is used to calculate the area of the triangle by integrating over the boundary of the triangle. This means that the line integral around the triangle's boundary is equal to the double integral over the triangle's interior.

3. What is the relationship between Green's Theorem and the divergence theorem?

The divergence theorem is a generalization of Green's Theorem in three dimensions. It states that the flux of a vector field through a closed surface is equal to the volume integral of the divergence of the vector field over the region enclosed by the surface. Green's Theorem can be seen as a special case of the divergence theorem in two dimensions.

4. How is Green's Theorem used in practical applications?

Green's Theorem is used in many practical applications, such as in physics to calculate work done by a force on a moving particle, in fluid dynamics to calculate the circulation of a fluid around a closed path, and in engineering for finding the center of mass of a two-dimensional object.

5. What are the advantages of using Green's Theorem over other methods?

Green's Theorem is often preferred over other methods for calculating line integrals because it transforms the line integral into a double integral, which can be easier to evaluate. Additionally, it is a powerful tool for solving problems in vector calculus and has many practical applications in various fields of science and engineering.

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