Greens theorem.

Green's theorem states that if F = <P,Q> has continuous partial derivatives on an open set containing the positively oriented simple closed curve C  and the region D it bounds, then the line integral of the the tangential component of F with repect to arc length around C is the double integral over D of the difference   .

The left hand side of Green's theorem has a nice meaning:  If the integral is positive, then on average  the tangential component of F is positive, so if we take the view that  F is a velocity flow for a fluid, that says that the net flow of the  fluid on the boundary curve C is counterclockwise.  If we divide by the length of the curve, we would get average signed speed that a point on the curve is rolled around the curve by the velocity field.  If the curve is a small circle of radius r then Green's theorem says that this average angular velocity is approximately one half the above difference of partial derivatives.  For this reason the the difference is a measure of the rotational tendency of the velocity field at each point, and as such is called the curl of F.    

Note :  The curl of a 3 dimensional velocity field F = <P, Q, R> is the vector   Del  X  F = < > .  If F is 2 dimensional (R = 0)  we can call the 3rd  component the curl of F.    Stokes theorem is a generalization of Greens theorem to the situation in space where C is a space curve and D is a surface bounded by C.  It says that the line integral of  the tangential component of F around C is equal to the surface integral of the normal component of curl F over D.

If instead of integrating the tangential component of the velocity field F around the curve C, we integrate the normal component of F, we get another line integral which can also be equated to a double integral by Green's theorem.  The outward normal n to the curve C is  obtained by rotating the  tangent vector <dx/dt, dy/dt>/|r'(t)|   90 degrees clockwise  to get  n =  <dy/dt, -dx/dt>/|r'(t)|

The line integral of the normal component of F = <P,Q> then becomes .  By Greens theorem this is equal to the the double integral   ,   This is called the normal form of Greens theorem.   The right hand side of this equation measures the transport of fliud across the curve  C (positive means fluid is flowing out on average, negative means fluid is flowing in).   The integrand of the right hand integral is  called the divergence of the velocity field F.   More generally, the divergence of a vector field F = <P,Q,R> is  defined as div F = del dot F = .    The Divergence theorem  is a generalization of  the normal form of Green's theorem to the situation in space where  you have a surface R bounding a soliid S in space where there is a nice velocity field F defined on an open set containing S and R.  It says that the surface integral of the (outward) normal component of  F over R is equal to the triple integral of the divergence of F over the solid S.

procedure 1 to investigate velocity fields: Tangential form of Greens theorem.

integration review

Integration review:

Defintions:  Know these and be able to use them.  

Riemann sum

Double integral

Area and volume as a double integral

Iterated integral

Type 1 and type 2 regions

Average value of a function on a region

first and second moments (about a line) of a function on a region

center of mass of a lamina, centroid of a region

Radius of gyration of a lamina about the y-axis (x-axis)

Triple integral

Volume as a triple integral

Line integrals with respect to arc length, x, y, or z (if the curve is a space curve)

Length as a line integral

vector field  Force field

Work as a line integral

gradient field

conservative vector field

potential

Theorems:  Know these and their consequences

Linearity of double integrals (and triple integrals and line integrals).

Comparability of double integrals ( and triple integrals and line integrals)

Fubini's Theorem

Change of variables Theorem

Fundamental Theorem for line integrals

Characterization Theorem for conservative vector fields

Green's Theorem

Typical problems

16_12
Compute a Riemann sum for a function
Compute a Riemann sum for a table of values
Double integral as the limit of a Riemann sum.
Compute a double integral by realizing it as a volume
Use Fubini's Theorem to compute a double integral
Compute an iterated integral
Compute a volume using an iterated integral
Compute an average height
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16_34
double integral trivia
compute a double integral over a type 1 or type 2
another compute a double integral over a type 1 or type 2
another compute a double integral over a type 1 or type 2
reverse the order of integration
convert to polar coordinates
compute a volume using polar coordinates
What's left after a hole is drilled in a sphere?
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16_56
compute the center of mass of a triangular region with variable density
Find center of mass of quarter disk
Find radius of gyration of rectangle in first quadrant
compute the area of a plane which lies above a given triangle.
another compute a double integral over a type 1 or type 2
compute an area using polar coordinates
What's left after a hole is drilled in a sphere?
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16_78
evaluate a triple iterated integral
compute an average temperature
compute an average temperature using cylindrical coordinates
compute a mass using spherical coordinates
y-coordinate of the center of mass.
Find average density of a prism
change of variable problem for double integrals
change of variable problem for triple integrals

17_12
identify a gradient field
Find the area of a fence.  Find its average height
Find the mass and center of mass of a wire  Find its average density.
Calculate line integrals with respect to x or y
Calculate a work integral along various paths from A to B
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17_34
Calculate line integrals for gradient fields
Find potentials.

Calculate work integral using Green's theorem

Express area as line integral and evaluate.

Sample questions.  More later in week .

1.   Draw a picture of the region R for the integral    .  Then reverse the order of integration and evaluate.

2.   Find the average value of      over the  rectangular solid with opposite corners  (0,0,0) and (1,2,3)

3.  Let a > 0. Let  be the solid below the plane , above the xy-plane and inside the cylinder .  Express the volume of  as an iterated double integral in polar coordinates.  Evaluate. Calculate the centroid of .

4.  Find the maximum and minimum values M and m of  on the domain D = {(x,y)|   }.   Then show that  the integral of f over D  is between    and

5.  Find the center of mass of the line segment  from (0,1,0) to (2,0,1)  if the density at each point on the wire is given by .

6.  Show that the vector field  F =<P,Q> = < >  is conservative.   Use this to evaluate  the work integral   .  over any nice curve C from (0,0) to (1,1).

7. Obtain the formula for the area of a triangle with base b and height h by evaluating a line integral