Green’s Theorem

Green’s theorem in a plane is a special case of Stroke’ theorem. It states “ If a closed region in XY-plane bounded by a closed curve and P and Q are continuous functions of x and Y having continuous derivatives then

$$ \oint_c Pdx + \oint_ Qdy = \iint _s \left (\frac {\partial Q}{\partial x} - \frac {\partial P}{\partial y}\right )dxdy$$

where the curve is traversed in anticlockwise direction. ”

Proof

Let us consider any well behaved vector field \(\vec A\) defined in the volume V bounded by a closed surface S. let \(\vec A = \phi \nabla \psi \) where \(\phi \) and \(\psi \) be any scalar fields.

Now the divergence theorem states that

\begin{align*} \int _v (\vec {\nabla} . \vec A) d^3 x &= \oint _s \vec A. \widehat n da \dots (1) \\ \text {We know} \\ \nabla . \vec A &= \left (\hat i\frac {\partial }{\partial x}+ \hat j \frac {\partial}{\partial y}+ \hat k\frac {\partial }{\partial z} \right ). (\hat I A_x + \hat j A_y + \hat k A_z) \\ &= \frac {\partial A_x }{\partial x} + \frac {\partial A_y}{\partial y} + \frac {\partial A_z}{\partial z} \dots (2) \\ \text {where}\: A &= \phi \nabla \psi \\ \end{align*}

\begin{align*}\hat i A_x + \hat j A_y + \hat k A_z &= \phi \left [(\hat i\frac {\partial \psi}{\partial x}+ \hat j \frac {\partial \psi}{\partial y}+ \hat k\frac {\partial \psi}{\partial z} \right ] \\ \text {Then}\: A_x &= \phi\frac {\partial \psi}{\partial x} ,\: A_y = \phi \frac {\partial \psi}{\partial y} \: \text {and}\: A_z = \frac {\partial \psi}{\partial z}, \\ \end{align*}

\begin{align*} \text {Then from}\: (2),\\ \nabla .\vec A &= \frac {\partial}{\partial x}\left (\phi \frac {\partial \psi}{\partial x}\right ) + \frac {\partial}{\partial y}\left (\phi \frac {\partial \psi}{\partial y}\right ) + \frac {\partial}{\partial z}\left (\phi \frac {\partial \psi}{\partial z}\right ) \\ &= \phi \frac {\partial ^2\psi}{\partial x^2} + \frac {\partial \phi}{\partial x}. \frac {\partial \psi}{\partial x} + \phi \frac {\partial ^2\psi}{\partial y^2} + \frac {\partial \phi}{\partial y} .\frac {\partial \psi}{\partial y} + \phi \frac {\partial ^2\psi}{\partial z^2} + \frac {\partial \phi}{\partial z} .\frac {\partial \psi}{\partial z}\\ &=\phi\left (\hat i\frac {\partial^2 \psi}{\partial x^2}+ \hat j \frac {\partial ^2\psi}{\partial y^2}+ \hat k\frac {\partial ^2\psi}{\partial z^2} \right )\end{align*}

\begin{align*} &=\phi\left (\hat i\frac {\partial^2 \psi}{\partial x^2}+ \hat j \frac {\partial ^2\psi}{\partial y^2}+ \hat k\frac {\partial ^2\psi}{\partial z^2} \right )+\frac {\partial \phi}{\partial x}. \frac {\partial \psi}{\partial x} + \frac {\partial \phi}{\partial y} .\frac {\partial \psi}{\partial y} + \phi \frac {\partial ^2\psi}{\partial z^2} + \frac {\partial \phi}{\partial z} .\frac {\partial \psi}{\partial z} \\ \nabla .(\phi\nabla\psi) &= \phi \nabla ^2\psi + \nabla\phi .\nabla \psi \dots (3)\\\text {and} \: \phi \nabla \psi . \: \widehat n &= \psi \frac {\partial \psi}{\partial n}\dots (4)\end{align*}

Where \(\frac {\partial}{\partial n}\) is the normal derivative at the surface S (directed outward from inside the volume V). When substituting (3) and (4) in the equation (1), we get

$$\int\limits_v (\phi \nabla ^2 \psi + \nabla \phi.\nabla\psi)d^3x = \oint\limits_s \phi \frac {\partial \psi}{\partial n}da \:\dots (5)$$

which is Green’s first identity. Now on interchanging \(\phi \) and \(\psi \) and then subtract it from equation (5) then we get

$$\int\limits_v (\phi \nabla ^2 \psi - \psi \nabla ^2\psi)d^3x = \oint\limits_s \left [\phi \frac {\partial \psi}{\partial n} - \psi \frac {\partial \phi}{\partial n}\right ]da $$

which is Green’s second identity.

Some important formula

If \(\phi \) and \(\psi \) are scalar. \(\vec A,\: \vec B,\: \vec C\) are vector quantities then

1. \( \nabla (\phi + \psi) = \nabla \phi + \nabla \psi\)
2. \(\nabla (\phi \psi) = \phi\nabla \psi + \psi \nabla \phi \)
3. \(\text {Div} (\vec A + \vec B )= \text {Div} (\vec A )+ \text {Div} (\vec B ) \)
4. \(\text {Curl} (\vec A + \vec B )= \text {Curl} (\vec A )+ \text {Curl} (\vec B ) \)
5. \(\nabla (\vec A. \vec B) = \vec A \times (\vec {\nabla} \times \vec B) + (\vec A. \vec {\nabla }) \vec B + \vec B \times (\vec {\nabla} \times \vec A) + (\vec B. \vec {\nabla })\vec A\)

6. \(\text {div}\: (\phi \:\vec A) = \phi\: \text {div}\: \vec A + \vec A\: \text {grad}\: \phi\)
7. \(\text {curl}\: (\phi \:\vec A) = \phi\: \text {curl}\: \vec A + \text {grad}\:\phi\times \vec A\)
8. \(\text {div curl} \: \vec A = 0\)
9. \(\text {curl grad}\: \phi = 0\)
10. \(\text {div}(\vec A \times \vec B) = \vec B.\: \text {curl}\: \vec A - \vec A\: \text {curl}\: \vec B\)

Bibliography

P.B. Adhikari, Bhoj Raj Gautam, Lekha Nath Adhikari. A Textbook of Physics. kathmandu: Sukunda Pustak Bhawan, 2011.

Jha, V. K.; 'Lecture title'; Elementary Vector Analysis; St. Xavier's College, Kathmandu; 2016.