Stroke's Theorem

Stroke’s theorem states that, “If \(\vec F\) and its derivative are continuous, the line integral of \(\vec F\) around a closed curve I is equal to the normal surface integral of curl \(\vec F\) over an open surface bounded by l.”

Simply, the surface integral of curl of \(\vec F\) taken over any open surface S is equal to the line integral of \(\vec F\) around periphery l of the surface.

In equation form, it can be written as,

$$\oint _l \vec F. \vec {dl} = \iint_s \nabla \times \vec F. \vec {dS}, \times (1) $$

Proof

Right hand side of the equation (1)

\begin{align*} \iint_s \nabla \times \vec F. \vec{dS} &= \int _s\nabla \times \vec F.\widehat n dS \\ &= \ii\nt _s \widehat {n}[\nabla \times \hat i F_x + \nabla \times jF_y + \nabla \times \hat k F_z]]dS, \dots (2) \end{align*}

The first integral of right hand side in equation (2) reduces to

\begin{align*}\iint_s \widehat n (\nabla \times \hat iF_x)dS &= \iint _s \widehat n.\left ( \left | \begin{matrix} \hat i &\hat j &\hat k \\\frac {\partial}{\partial x} &\frac {\partial}{\partial y} &\frac {\partial}{\partial z} \\F_x &0 &0 \end{matrix}\right|\right )dS \\ &= \iint_s\left[\widehat n. \hat j\frac {\partial F_x}{\partial z} - \widehat n. \hat k\frac {\partial F_x}{\partial y} \right ]dS, \dots (3)\end{align*}

Here, the projection of dS onto the xy-plane leads to

$$ \widehat n. \hat k dS = dx\: dy, \dots (4) $$

Now, let the line segment P₁P₂ be the intersection of the surface S with a plane that is parallel to the yz-plane at a distance x from the origin. Along the strip P₁P₂ it can be written

\begin{align*} dF_x &= \frac {\partial F_x}{\partial y}dy + \frac {\partial F_x}{\partial z}dz, \dots (5)\\ \text {Here}\\ \d\vec R &= dy\hat j + dz \hat k \\ \end{align*}

Since, the vector \(d\vec R\) is tangent to P₁P₂ at point A.

\begin{align*} d\vec R. \widehat n &= dy\:\widehat n.\hat j + dz \widehat n \hat k \\ \text {or,}\: \vec n. \hat j &= -\frac {dz}{dy}\:\widehat n. \hat k \\ &= - \frac {dz}{dy}\left (\frac {dxdy}{dS}\right ) \\ \text {or,}\:\widehat n. \hat j dS &= -dx\: dz, \dots (6) \end{align*}

Using equation (4), (5) and (6) in equation (3), becomes

\begin{align*}\iint _s \widehat n.\: (\nabla \times \hat I F_x)dS &= -\iint _s \left [ \frac {\partial F_x}{\partial z}dz + \frac {\partial F_x}{\partial y}dy\right ]dx \\&= - \int [F_x(x_2, y_2, z_2) - F_x (x_1, y_1, z_1)]dx, \dots (7) \end{align*}

The sense of the periphery at P₁ is positive, \(dx = dl_x\), where as it is negative at P₂, \(dx = -dl_x\) now equation (7) becomes

\begin{align*} \iint _s \widehat n (\nabla \times \hat I F_x)dS &= \int F_x(x_2, y_2, z_2)dl_x +\int F_x (x_1, y_1, z_1)]dl_x \\ &= \oint _l F_xdl_x \dots (8)\end{align*}

The second and third integrals of right hand side of equation (2) reduce to

\begin{align*} \iint _s \widehat n (\nabla \times \hat j\: F_y)dS &= \oint _lF_ydly, \dots (9) \\ \text {also,}\\ \iint _s \widehat n (\nabla \times \hat k\: F_z)dS &= \oint _lF_zdlz, \dots (10) \\ \text {From equations}\: (8),\: (9),\text {and}\: (10),\: \text {it becomes} \\\int _s \nabla \times \vec F. \vec {dS} &= \oint _l F_xdl_x + \oint_l F_ydl_y + \oint_l F_zdl_z \\\iint \nabla \times \vec F. \vec {dS} &= \oint \vec F. \vec {dl}, \dots (11)\end{align*}

Equation (11) shows Stroke’s theorem. The Stroke’s theorem is extremely useful in potential theory as well as in other areas of mathematical physics.

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.