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Mathematics LibreTexts

4.8: Green’s Theorem in the Plane

Green's Theorem, allows us to convert the line integral into a double integral over the region enclosed by \(C\). The discussion is given in terms of velocity fields of fluid flows (a fluid is a liquid or a gas) because they are easy to visualize. However, Green's Theorem applies to any vector field, independent of any particular interpretation of the field, provided the assumptions of the theorem are satisfied. We introduce two new ideas for Green's Theorem: divergence and circulation density around an axis perpendicular to the plane. 

Divergence

Suppose that \(F(x, y) = M(x, y) \hat{\textbf{i}} + N(x, y) \hat{\text{j}}\),  is the velocity field of a fluid flowing in the plane and that the first partial derivatives of \(M\) and \(N\) are continuous at each point of a region \(R\).

Let \((x, y)\) be a point in \(R\) and let \(A\) be a small rectangle with one corner at \((x, y)\) that, along with its interior, lies entirely in \(R\). The sides of the rectangle, parallel to the coordinate axes, have lengths of \( \Delta x \) and \( \Delta y \). Assume that the components \(M\) and \(N\) do not change sign troughout a small region containing the rectangle \(A\). The rate at which fluid leaves the rectangle across the bottom edge is approximately 

\[F(x,y)=M(x,y) \hat{\textbf{i}}+N(x,y) \hat{\textbf{j}}\]

This is the scalar component of the velocity at \((x,y)\) in the direction of the outward normal times the length of the segment. If the velocity is in meters per second, for example, the flow rate will be in meters per second times meters or square meters per second. The rates at which the fluid crosses the other three sides in the directions of their outward normals can be estimated in a similar way. The flow rates may be positive or negative depending on the signs of the components of \(F\). We approximate the net flow rate across the rectangular boundary of \(A\) by summing the flow rates across the four edges as defined by the following dot products. 

  • Top:  \[F(x,y+\Delta y)\cdot (\hat{\textbf{j}})\Delta x=-N(x,y+\Delta y)\Delta x\]
  • Bottom:  \[F(x,y)\cdot (-\hat{\textbf{j}})\Delta x=-N(x,y)\Delta x\]
  • Right:  \[F(x+\Delta x,y)\cdot (\hat{\textbf{i}})\Delta y=M(x+\Delta x,y)\Delta y\]
  • Left:  \[F(x,y)\cdot (-\hat{\textbf{i}})\Delta y=-M(x,y)\Delta y\]

Summing opposite pairs gives 

Top and bottom: \[(N(x,y+\Delta y)-N(x,y))\cdot(\Delta x)\]

Right and left:  \[(M(x+\Delta x,y)-M(x,y))\cdot(\Delta y)\]

Adding these last two equations gives the net effect of the flow rates, or the Flux across rectangle boundary.

We now divide by \(xy\) to estimate the total flux per unit area or flux density for the rectangle:

Finally, we let and Jly approach zero to define the flux density of \(F\) at the point \((x,y)\). In mathematics, we call the flux density the divergence of \(F\). The symbol for it is div \(F\), pronounced "divergence of \(F\)' or "div \(F\)." 

The divergence (flux density) of a vector field \(F= \text{the point } (x,y)\) is

\[divF=\dfrac{\partial M}{\partial x}+\dfrac{\partial N}{\partial x}.\]

Spin Around an Axis: The k-Component of Curl 

The second idea we need for Green's Theorem bas to do with measuring how a floating paddle wheel, with axis perpendicular to the plane, spins at a point in a fluid flowing in a plane region. This idea gives some sense of how the fluid is circulating around axes located at different points and perpendicular to the region. Physicists sometimes refer to this as the circulation density of a vector field \(F\) at a point. To obtain it, we return to the velocity field

\[F(x,y)=M(x,y)\hat{\textbf{i}}+N(x,y)\hat{\textbf{j}}\]

and consider the rectangle \(A\) in Figure 16.29 (where we assume both components of \(F\) are positive). 

The circulation rate of \(F\) around the boundary of \(A\) is the sum of flow rates along the sides in the tangential direction. For the bottom edge, the flow rate is approximately

\[F(x,y)\cdot ( \hat{\textbf{i}} )\Delta x=-M(x,y)\Delta x\]

This is the scalar component of the velocity \(F(x, y)\) in the tangent direction \( \hat{\textbf{i}} \) times the length of the segment. The flow rates may be positive or negative depending on the components of \(F\). We approximate the net circulation rate around the rectangular boundary of \(A\) by summing the flow rates along the four edges as determined by the following dot products. 

  • Top: \[F(x,y + \Delta y) \cdot (-i) \Delta x= -M(x,y+ \Delta y)\Delta x \]
  • Bottom: \[F(x,y) \cdot ( \hat{\textbf{i}}) \Delta x = M(x,y) \Delta x\]
  • Right: \[F(x+\Delta x ,y ) \cdot (\hat{\textbf{j}} ) \Delta y = N(x+ \Delta x, y) \Delta y \]
  • Left: \[ F(x,y) \cdot (- \hat{\textbf{j}} \Delta y = - N(x,y) \Delta y \]

 

  • Top and bottom: \[-(M(x,y+\Delta y)-M(x,y))\cdot(\Delta x)\]
  • Right and left:  \[(N(x+\Delta x,y)-N(x,y))\cdot(\Delta y)\]

Adding these last two equations gives the net circulation relative to the counterclockwise orientation, and dividing by JlxJly gives an estimate of the circulation density for the rectangle:

Circulation around rectangle rectangle area

We let Jlx and Jly approach zero to define the circulation density of \(F\) at the point \((x,y)\).

If we see a counterclockwise rotation looking downward onto the xy-plane from the tip of the unit \( \hat{\textbf{k}}\) vector, then the circulation density is positive (Figure 16.30). The value of the circulation density is the \( \hat{\textbf{k}}\)-component of a more general circulation vector field we addressed in Section 16.7, called the curl of the vector field \(F\). For Green's Theorem, we need only this \( \hat{\textbf{k}}\) -component. 

The circulation density of a vector field \(F= M \hat{\textbf{i}} + N \hat{\textbf{j}}\) at the point \( ( x, y ) \)  is the scalar expression

\[\dfrac{\partial M}{\partial x} - \dfrac{\partial N}{\partial x} \]

Theorem 4.1: Green's Theorem (Flux-Divergence Form)

Let \(C\) be a piecewise smooth, simple closed curve enclosing a region \(R\) in the plane. Let \(F = M \hat{\textbf{i}} + N \hat{\textbf{j}}\) be a vector field with \(M\) and \(N\) having continuous first partial derivatives in an open region containing \(R\). Then the outward flux of \(F\) across \(C\) equals the double integral of \(div F\) over the region \(R\) enclosed by \(C\).

\[\oint_C F\cdot nds=\oint_CMdy-Ndx=\iint_{R}^{ } (\dfrac{\partial M}{\partial x}+\dfrac{\partial N}{\partial x})dxdy\]

Theorem 4.2: Green's Theorem (Circulation-Curl Form)

Let \(C\) be a piecewise smooth, simple closed curve enclosing a region \(R\) in the plane. Let \(F = M \hat{\textbf{i}} + N \hat{\textbf{j}}\) be a vector field with \(M\) and \(N\) having continuous first partial derivatives in an open region containing \(R\). Then the counterclockwise circulation of \(F\) around \(C\) equals the double integral of \((curl F) \cdot k\) over \(R\).

\[\oint_C F\cdot Tds=\oint_CMdy+Ndx=\iint_{R}^{ } (\dfrac{\partial N}{\partial x}-\dfrac{\partial M}{\partial x})dxdy\]

  • Integrated by Justin Marshall.