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George Green (17931841) is somewhat of an anomaly in mathematics. He was the son of a baker/miller in a rural area. He had only one year of formal education. For the rest he was selftaught, yet he discovered major elements of mathematical physics. One of those elements was a theorem, now known as Green's Theorem.
Let P(x,y) and Q(x,y) be differentiable functions defined over a region S. Let C be the piecewise smooth boundary of S.
The curve C is traversed in a counterclockwise direction so that the region S is always to the left of the direction of travel.
Then
The proof will be carried out using successively more general condition on the boundary C. First it is assumed that the region S has no holes. Second, it is assumed that C is vertically simple and horizontally simple.
Vertically simple means that the lower edge of C can be expressed as a single valued function of x; i.e., y=f_{1}(x). Call this the curve C_{1}. Likewise the upper edge of C can be expressed a single valued function of x; i.e., y=f_{2}(x). This curve will be called C_{2}. The functioncurve C_{1} runs from (x_{1} , y_{1}) to (x_{2}, y_{2}), but C_{2} runs in the reverse direction from (x_{2}, y_{2}) to (x_{1}, y_{1}).
Then
Similarly horizontally simple means that the left edge of C can be expressed x=g_{1}(y) and the right edge as x=g_{2}(y). These will be denoted as the curves C_{3} and C_{4}, respectively. The functioncurve C_{3} runs from (x_{3}, y_{3}) to (x_{4}, y_{4}) and C_{4} in the reverse direction (x_{4}, y_{4}) to (x_{3}, y_{3}).
Note the bit of asymmetry to this formulation. In traversing the curve C starting at (x_{1}, y_{1}) the lower (minimum) edge of S is first encountered. On the other hand, starting at (x_{3}, y_{3}) the rightmost (maximum) edge of S is first encountered.
Consider
But a reversal of the direction of integration changes the sign of the integral; i.e.,
Thus
Now consider ∫_{S}∫(∂P/∂y)dydx.
The inner integral with respect to y can be evaluated so
This latter expression is just the negative of the expression found above.
Therefore
Similarly, working with the functions g_{1} and g_{2} it can and will be shown below that
Thus
The proof that ∫_{C}Qdy = ∫_{S}∫(∂Q/∂x)dxdy starts with
The integration from (x_{4}, y_{4}) to (x_{3}, y_{3}) is just the negative of the integration from (x_{3}, y_{3}) to (x_{4}, y_{4}).
Thus
Therefore
Now consider
The inner integral evaluates to
Thus
This is exactly the expression found previously for ∫_{C}Qdy. Therefore
When the two propositions proved above are combined it can be concluded that
Consider a region with a hole in it, such as shown below
The boundary of this region consists of the outer boundary curve C' and the boundary curve of the hole, C". Traversing the boundary of S now consists of a counterclockwise traversal of the outer boundary and a clockwise traversal of the boundary of the hole. This procedure keeps the region S always to the left of the direction of travel.
Green's Theorem can be applied to a region with holes by cutting lines from the outer boundary to each hole, such as shown below.
This creates a region without holes. But the cut lines are traversed twice into the hole and out of the holes, as two parts of the boundary of the created region, as shown by c_{1} and c_{2}. The effects on the line integral of c_{1} and c_{2} cancel out. They enclose no area so there is no effect on the surface integral. Thus Green's Theorem applies to regions with holes.
(To be continued.)
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