Green’s Theorem
Green's theorem gives a relationship between the line integral of a vector field over a closed path and the double integral over the area it encloses. Along with the case of computing line integrals over conservative fields, Green's theorem provides an equivalence that makes for a quicker computation.
Green's theorem applies to any vector field, generalizing the method to formulate the line integrals. We will understand the significance of two new terms divergence and circulation density, which develop the idea of Green's theorem. If you are wondering why the name of this theorem is Green, it's because the name of the person who formulated this theorem was George Green.
Divergence and circulation density
Main wiki: divergence and circulation density.
Divergence
The divergence of a three-dimensional vector field is the extent to which the vector field flow behaves like a source at a given point. It is a local measure of its "out-going-ness"–-the extent to which there is more exiting an infinitesimal region of space than entering it. If the divergence is nonzero at some point, then there must be a source or sink at that position. ^{[1]}
To calculate the divergence at a given point, let us say we have an infinitesimally small rectangular region into which some amount of fluid flows. We calculate the fluid flow rates inside the rectangle and then we take the limits of the area of the rectangle to \(0\), so that we can get the divergence for a single point.
Let us say we have a velocity field which depicts the fluid flowing into a rectangular region (as in the image), and is defined by the following expression:
\[\mathbf F(x,y)=F_x(x,y)\hat{\mathbf x}+F_y\hat{\mathbf y}.\]
As in the image, we have the length and breadth of the rectangle to be \(\Delta x\) and \(\Delta y\). Thus, the fluid flow rates leaving a particular side of the rectangle will be
\[(\text{Velocity of the fluid at the respective corner})\times (\text{length of the side}),\]
according to which, the fluid flow rates at each of the sides of the rectangle would be
\[\begin{align} \text{Left} : \quad &\mathbf F(x,y)\cdot (-\hat{\mathbf x})\Delta y \quad\quad =-F_x(x,y)\Delta y\\ \text{Right} : \quad &\mathbf F(x+\Delta x,y)\cdot \hat{\mathbf x}\Delta y \quad =F_x(x+\Delta x,y)\Delta y\\ \text{Top} : \quad &\mathbf F(x,y+\Delta y)\cdot \hat{\mathbf y}\Delta x \quad =F_y(x,y+\Delta y)\Delta x\\ \text{Bottom} : \quad &\mathbf F(x,y)\cdot (-\hat{\mathbf y})\Delta x \quad\quad =-F_y(x,y)\Delta x. \end{align}\]
Now to find the whole outflow, we need to sum up the fluid flow rates at each of the sides of the rectangle, and we arrive at
\[\begin{align} \text{Left + Right} : \quad &-F_x(x,y)\Delta y + F_x(x+\Delta x,y)\Delta y = \left(\dfrac{\partial F_x}{\partial x}\Delta x\right)\Delta y\\ \text{Top + Bottom} : \quad &F_y(x,y+\Delta y)\Delta x -F_y(x,y)\Delta x \quad = \left(\dfrac{\partial F_y}{\partial y}\Delta y\right)\Delta x. \end{align}\].
Finally, adding the resulting values and dividing this sum by the area of the rectangle yields us the outflow per unit area (or flux density) which is nothing but the divergence of the fluid:
\[\dfrac{\left(\dfrac{\partial F_x}{\partial x}\Delta x\right)\Delta y + \left(\dfrac{\partial F_y}{\partial y}\Delta y\right)\Delta x}{\Delta x \Delta y}=\left(\dfrac{\partial F_x}{\partial x}\right) + \left(\dfrac{\partial F_y}{\partial y}\right).\]
The divergence is the flux density of a vector field \(\mathbf F = F_x\hat{\mathbf x} + F_y\hat{\mathbf y}\) at the points \(x\) and \(y,\) given by
\[\text{div }\mathbf F = \dfrac{\partial F_x}{\partial x}+\dfrac{\partial F_y}{\partial y}.\]
Circulation density
Now, we devise a method to find the rate at which the fluid is circulating along the region, and again we consider the case of a rectangle with the same velocity vector;
\[\mathbf F(x,y)=F_x(x,y)\hat{\mathbf x}+F_y\hat{\mathbf y}.\]
The rate of circulation of the velocity vector around the rectangle can be computed by adding the fluid flow rates at each side of the rectangle. This can be formulated by finding the scalar component of the velocity vector in the tangential direction and multiplying it by the respective side length.
Thus, we arrive at the circulation rates along each side of the rectangle:
\[\begin{align} \text{Left} : \quad &\mathbf F(x,y)\cdot (-\hat{\mathbf y})\Delta y \quad\quad\quad =-F_y(x,y)\Delta y\\ \text{Right} : \quad &\mathbf F(x+\Delta x,y)\cdot \hat{\mathbf y}\Delta y \quad\quad =F_y(x+\Delta x,y)\Delta y\\ \text{Top} : \quad &\mathbf F(x,y+\Delta y)\cdot (-\hat{\mathbf x})\Delta x \ \ =-F_x(x,y+\Delta y)\Delta x\\ \text{Bottom} : \quad &\mathbf F(x,y)\cdot \hat{\mathbf x}\Delta x \quad\quad\quad\quad \ \ =F_x(x,y)\Delta x, \end{align}\]
Adding the opposite pairs as we did in the divergence vector, we get
\[\begin{align} \text{Left + Right} : \quad &-F_y(x,y)\Delta y + F_y(x+\Delta x,y)\Delta y = \left(\dfrac{\partial F_y}{\partial x}\Delta x\right)\Delta y\\ \text{Top + Bottom} : \quad &-F_x(x,y+\Delta y)\Delta x -F_x(x,y)\Delta x \quad = -\left(\dfrac{\partial F_x}{\partial y}\Delta x\right)\Delta y. \end{align}\]
Adding the result and dividing it by the area of the rectangle, we get the circulation density of the fluid flowing into the rectangular region:
\[\dfrac{\left(\dfrac{\partial F_y}{\partial x}\Delta x\right)\Delta y - \left(\dfrac{\partial F_x}{\partial y}\Delta x\right)\Delta y}{\Delta x \Delta y}=\left(\dfrac{\partial F_y}{\partial x}\right) -\left(\dfrac{\partial F_x}{\partial y} \right).\]
The circulation density of a vector field \(\mathbf F = F_x\hat{\mathbf x} + F_y\hat{\mathbf y}\) at the points \(x\) and \(y\) is given by
\[(\text{curl }\mathbf F)\cdot \hat{\mathbf z}= \dfrac{\partial F_x}{\partial x}-\dfrac{\partial F_y}{\partial y}.\]
The expression is the \(\mathbf z\) component of the curl vector. For more information on the curl vector, visit Stokes' theorem.
Green's theorem
Based on the definition of divergence and circulation density, there can be two forms of the Green's theorem.
Let \(C\) be a piecewise smooth, simple closed curve enclosing a region \(R\) in the two-dimensional plane. Let there be a vector field \(\mathbf F = F_x\hat{\mathbf x} + F_y\hat{\mathbf y}\) with \(F_x\) and \(F_y\) having continuous first partial derivatives which is present inside \(R\). Then, the line integral of \(\mathbf F\) can be computed in two different ways.^{[2]}1. The outward flux of \(\mathbf F\) across \(C\), giving us the Divergence integral:
\[\oint_C \mathbf F\cdot d\mathbf s = \iint_R \text{div }\mathbf F = \underbrace{\iint_R \left(\dfrac{\partial F_y}{\partial x}+\dfrac{\partial F_x}{\partial y}\right) \, dx \, dy}_{\text{Divergence Integral}}.\]
2. The counter-clockwise circulation of \(\mathbf F\) around the region \(C\), giving us the curl integral:
\[\oint_C \mathbf F\cdot d\mathbf s = \iint_R (\text{curl }\mathbf F)\cdot \mathbf k =\underbrace{\iint_R \left(\dfrac{\partial F_y}{\partial x}-\dfrac{\partial F_x}{\partial y}\right) \, dx \, dy}_{\text{Curl Integral}}.\]
[Lots of examples]
Proof
Green's theorem is limited to two dimensions and so is the proof. Stokes generalized this theorem and formulated Stokes' theorem for higher dimensions. But in this wiki, we will just prove Green's theorem, with the proof being two-fold.
This proof is the reversed version of another proof; watch it here.
Let \(C\) be a piecewise smooth, simple closed curve in the plane. Let this smooth curve be enclosed in the region \(R\), and assume that \(F_x\) and \(F_y\) and their first partial derivatives are continuous at each point in the region \(R\) containing \(C\). We need to prove that
\[\oint_C \mathbf F\cdot d\mathbf s =\oint_C F_x \, d\hat{\mathbf x}+F_y \, d\hat{\mathbf y} =\iint_R \left(\dfrac{\partial F_y}{\partial x}-\dfrac{\partial F_x}{\partial y}\right) \, dx \, dy.\]
Let us say that the curve \(C\) is made up of two curves \(C_1\) and \(C_2\) such that
\[C_1: y = f_1(x) \ \forall a\leq x\leq b\\C_2: y = f_2(x) \ \forall b\leq x\leq a.\]
Now, under the following conditions, integrating \(\partial F_x/\partial y\) with respect to \(y\) between \(y=f_1(x)\) and \(y=f_2(x)\) yields
\[\int_{f_1(x)}^{f_2(x)}\dfrac{\partial F_x}{\partial y} \, dy = F_x(x,f_2(x))-F_x(x,f_1(x)).\]
Integrating the resulting integrand over the interval \((a,b)\) we obtain
\[\begin{align} \int_a^b\int_{f_1(x)}^{f_2(x)}\dfrac{\partial F_x}{\partial y} \, dy \, dx &=\int_a^b (F_x(x,f_2(x))-F_x(x,f_1(x))) \, dx\\ &=\int_a^b (F_x(x,f_2(x)) \, dx -\int_a^b (F_x(x,f_1(x)) \, dx\\ &=-\int_b^a (F_x(x,f_2(x)) \, dx - \int_a^b (F_x(x,f_1(x)) \, dx\\ &=-\int_{C_2} F_x \, dx-\int_{C_1} F_x \, dx \\ &=-\oint_{C} F_x \, dx.\\ \end{align}\]
Thus, we arrive at the first half of the required expression
\[\oint_{C} F_x \, dx = \int_a^b\int_{f_1(x)}^{f_2(x)}\dfrac{\partial F_x}{\partial y} \, dy \, dx=\iint_R \left(-\dfrac{\partial F_x}{\partial y}\right) \, dx \, dy.\]
Similarly, we can arrive at the other half of the proof. Let us say the curve \(C\) is made up of two curves \(C_1\) and \(C_2\) such that
\[C_1: y = g_1(x) \ \forall d\leq x\leq c\\C_2: y = g_2(x) \ \forall c\leq x\leq d.\]
Now, integrating \(\partial F_y/\partial x\) with respect to \(x\) between \(x=g_1(y)\) and \(x=g_2(y)\) yields
\[\int_{g_1(x)}^{g_2(x)}\dfrac{\partial F_y}{\partial x} \, dx = F_y(g_2(y),y)-F_y(g_1(x),y).\]
Integrating the resulting integrand over the interval \((c,d)\) we obtain
\[\begin{align} \int_c^d\int_{g_1(x)}^{g_2(x)}\dfrac{\partial F_y}{\partial x} \, dy \, dx &=\int_c^d (F_y(g_2(x),y)-F_y(g_1(x),y)) \, dx\\ &=\int_c^d (F_y(g_2(x),y) \, dy -\int_c^d (F_y(g_1(x),y) \, dy\\ &=-\int_d^c (F_y(g_2(x),y) \, dy - \int_c^d (F_y(g_1(x),y) \, dy\\ &=-\int_{C_2} F_y \, dy-\int_{C_1} F_y \, dy \\ &=-\oint_{C} F_y \, dy.\\ \end{align}\]
Thus, we arrive at the second half of the required expression. \[\oint_{C} F_y \, dy = \int_c^d\int_{g_1(x)}^{g_2(x)}\dfrac{\partial F_y}{\partial x} \, dy \, dx= \iint_R \left(\dfrac{\partial F_y}{\partial x}\right) \, dx \, dy.\]
Summing both the results finishes the proof of the Green's theorem, which is
\[\begin{align} \oint_{C} F_y \, dy+\oint_{C} F_y \, dy=\oint_C \mathbf F\cdot d\mathbf s &=\iint_R \left(-\dfrac{\partial F_x}{\partial y}\right) \, dx \, dy + \iint_R \left(\dfrac{\partial F_y}{\partial x}\right) \, dx \, dy\\ &=\iint_R \left(\dfrac{\partial F_y}{\partial x}-\dfrac{\partial F_x}{\partial y}\right) \, dx \, dy. \ _\square \end{align}.\]
See Also
References
- Jess Brewer, H. DIVERGENCE of a Vector Field. Retrieved from http://musr.phas.ubc.ca/~jess/hr/skept/Gradient/node4.html
- George B. Thomas, J. (2014). Thomas' Calculus. Pearson India Education Services.