Calculating \int F dr with Green's Theorem

In summary, Green's Theorem is a mathematical theorem used to relate the line integral of a two-dimensional vector field over a closed curve to a double integral over the region enclosed by the curve. It can be used to calculate line integrals by first expressing them as double integrals. The vector field in Green's Theorem represents the rate of change of a scalar field in a two-dimensional region. It can only be used for closed curves and has limitations such as only applying to two-dimensional spaces and conservative vector fields. If an open curve needs to be calculated, Stokes' Theorem can be used.
  • #1
bugatti79
794
1

Homework Statement



Use Green's Theorem to calculate [itex]\int F dr[/itex]

Homework Equations



[itex] F(x,y)= (\sqrt x +y^3) i + (x^2+ \sqrt y) j[/itex] where C is the arc of y=sin x from (0,0) to ( pi,0) followed by line from (pi,o) to (0,0).



The Attempt at a Solution



We have [itex]\int f dx + g dy = \int \int_R (g_x-f_y)[/itex] dA for counterclockwise rotation, but the question is given in clockwise rotation so does green's theorem become

[itex]- \int \int_R (g_x-f_y) dA[/itex]...? Ie, a sign change?
 
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  • #2
bugatti79 said:

Homework Statement



Use Green's Theorem to calculate [itex]\int F dr[/itex]

Homework Equations



[itex] F(x,y)= (\sqrt x +y^3) i + (x^2+ \sqrt y) j[/itex] where C is the arc of y=sin x from (0,0) to ( pi,0) followed by line from (pi,o) to (0,0).



The Attempt at a Solution



We have [itex]\int f dx + g dy = \int \int_R (g_x-f_y)[/itex] dA for counterclockwise rotation, but the question is given in clockwise rotation so does green's theorem become

[itex]- \int \int_R (g_x-f_y) dA[/itex]...? Ie, a sign change?

Any clues on this one?

Thanks
 
  • #3
bugatti79 said:

Homework Statement



Use Green's Theorem to calculate [itex]\int F dr[/itex]

Homework Equations



[itex] F(x,y)= (\sqrt x +y^3) i + (x^2+ \sqrt y) j[/itex] where C is the arc of y=sin x from (0,0) to ( pi,0) followed by line from (pi,o) to (0,0).

The Attempt at a Solution



We have [itex]\int f dx + g dy = \int \int_R (g_x-f_y)[/itex] dA for counterclockwise rotation, but the question is given in clockwise rotation so does green's theorem become

[itex]- \int \int_R (g_x-f_y) dA[/itex]...? Ie, a sign change?
Yes, clockwise gives the opposite sign compared to counter-clockwise.
 
  • #4
SammyS said:
Yes, clockwise gives the opposite sign compared to counter-clockwise.

Thanks
 
  • #5
bugatti79 said:

Homework Statement



Use Green's Theorem to calculate [itex]\int F dr[/itex]

Homework Equations



[itex] F(x,y)= (\sqrt x +y^3) i + (x^2+ \sqrt y) j[/itex] where C is the arc of y=sin x from (0,0) to ( pi,0) followed by line from (pi,o) to (0,0).



The Attempt at a Solution



We have [itex]\int f dx + g dy = \int \int_R (g_x-f_y)[/itex] dA for counterclockwise rotation, but the question is given in clockwise rotation so does green's theorem become

[itex]- \int \int_R (g_x-f_y) dA[/itex]...? Ie, a sign change?

Can anyone confirm this integral is set up correctly?

[itex]\displaystyle \int_0^ {\pi} \int_0^ {sin x} (3y^2-2x) dy dx[/itex]
 
  • #6
bugatti79 said:
Can anyone confirm this integral is set up correctly?

[itex]\displaystyle \int_0^ {\pi} \int_0^ {sin x} (3y^2-2x) dy dx[/itex]

That looks OK to me.
 

FAQ: Calculating \int F dr with Green's Theorem

1. What is Green's Theorem?

Green's Theorem is a mathematical theorem that relates the line integral of a two-dimensional vector field over a closed curve to a double integral over the region enclosed by the curve.

2. How is Green's Theorem used to calculate \int F dr?

In order to use Green's Theorem to calculate \int F dr, you must first express the line integral as a double integral using the formula \int F dr = \int\int_R (\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}) dA, where P and Q are the components of the vector field and R is the region enclosed by the curve.

3. What is the significance of the vector field in Green's Theorem?

The vector field in Green's Theorem represents the rate of change of a scalar field, such as temperature or pressure, in a two-dimensional region. It provides information about the direction and magnitude of change at each point.

4. Can Green's Theorem be used to calculate line integrals over open curves?

No, Green's Theorem can only be used to calculate line integrals over closed curves. If you need to calculate a line integral over an open curve, you can use a different theorem called Stokes' Theorem.

5. Are there any limitations to using Green's Theorem to calculate line integrals?

Yes, Green's Theorem can only be used in two-dimensional spaces and for conservative vector fields. It also requires the region enclosed by the curve to be simply connected (without any holes or gaps).

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