Thread: Integral field vector

Explain how to relate the integral curvilinear of field vector with the area the region limited by a regular curve, simple and closed ? Apply the result and calculate the area of the ellipse:

Consider Green's Theorem

For F = (x-y) i +(x +y) j

d(x+y)/dx = 1 d(x-y)/dy = -1

The line integral then is 2times the double integral over R , or twice the area of the region enclosed by your ellipse

It is easier to compute this with the line integral than with a double integral.

so the area enclosed is 1/2 the line integral

parameterize C with x = acos(t) y = bsin(t)

and now you are good to go.

Remember you should get pi*ab

Originally Posted by Calculus26

Consider Green's Theorem

For F = (x-y) i +(x +y) j

d(x+y)/dx = 1 d(x-y)/dy = -1

The line integral then is 2times the double integral over R , or twice the area of the region enclosed by your ellipse

It is easier to compute this with the line integral than with a double integral.

so the area enclosed is 1/2 the line integral

parameterize C with x = acos(t) y = bsin(t)

and now you are good to go.

Remember you should get pi*ab

I find because my integral the correct is ?

No-- remember the area is 1/2 the line integral--so 0 t0 2pi is correct

if you use 0 to pi you don't have a closed curve

Originally Posted by Calculus26

No-- remember the area is 1/2 the line integral--so 0 t0 2pi is correct

if you use 0 to pi you don't have a closed curve

Ok thank you the answer is

No the answer is pi *ab The line integral is 2pi*ab and the AREA is 1/2 this

Originally Posted by Calculus26

No the answer is pi *ab The line integral is 2pi*ab and the AREA is 1/2 this

why the AREA is ?

See my first post on this -- I explained it there

Originally Posted by Calculus26

See my first post on this -- I explained it there

Ok te green theorem is 2 AREA 1/2. If green theorem is 1 AREA 1, if 3 AREA 1/3 ??

That would make sense wouldn't it ?

If F = f i +g j

And if dg/dx- df/dy is a constant k then Green's theorem gives k *Area

Originally Posted by Calculus26

That would make sense wouldn't it ?

If F = f i +g j

And if dg/dx- df/dy is a constant k then Green's theorem gives k *Area

Yes thank you