SURFACE INTEGRALS | Improve Society

Fig. 6-3

Let $S$ be a two-sided surface having projection $\cal R$ on the $xy$ plane as in the adjoining Fig. 6-3. Assume that an equation for $S$ is $z = f(x, y)$ , where $f$ is single-valued and continuous for all $x$ and $y$ in $\cal R$ . Divide $\cal R$ into $n$ subregions of area $\Delta A_p,\ p = 1,\ 2,\ \dots,\ n$ , and erect a vertical column on each of these subregions to intersect $S$ in an area $\Delta S_p$ .

Let $\phi (x, y, z)$ be single-valued and continuous at all points of $S$ . Form the sum
$\displaystyle \sum_{p=1}^{n}\phi(\xi_p, \eta_p, \zeta_p)\Delta S_p \cdots(29)$
where $(\xi_p, \eta_p, \zeta_p)$ is some point of $\Delta S_p$ . If the limit of this sum as $n \rightarrow \infty$ in such a way that each $\Delta S_p \rightarrow 0$ exists, the resulting limit is called the surface integral of $\phi(x, y, z)$ over $S$ and is designated by
$\displaystyle \underset{S}{\iint}\phi(x, y, z)dS\cdots(30)$
Since $\Delta S_p = |\sec\gamma_p|\Delta A_p$ approximately, where $\gamma_p$ is the angle between the normal line to $S$ and the positive $z$ axis, the limit of the sum (29) can be written
$\displaystyle \underset{\cal R}{\iint}\phi(x, y, z)|\sec\gamma|dA\cdots(31)$
The quantity $|\sec\gamma|$ is given by
$\displaystyle |\sec\gamma| = \frac{1}{|\bold{n}_p\cdot\bold{k}|} = \sqrt{1 + \left( \frac{\partial z}{\partial x} \right)^2 + \left( \frac{\partial z}{\partial y} \right)^2}\cdots(32)$
Then assuming that $x = f(x, y)$ has continuous (or sectionally continuous) derivatives in $\cal R$ , (31) can be written in rectangular form as
$\displaystyle \underset{\cal R}{\iint}\phi(x, y, z)\sqrt{1 + \left( \frac{\partial z}{\partial x} \right)^2 + \left( \frac{\partial z}{\partial y} \right)^2}dxdy \cdots(33)$
In case the equation for $S$ is given as $F(x, y, z) = 0$ , (33) can also be written
$\displaystyle \underset{S}{\iint}\phi(x, y, z)\frac{\sqrt{(F_x)^2 + (F_y)^2 + (F_z)^2}}{|F_z|}dxdy\cdots(34)$
The results (33) or (34) can be used to evaluate (30).

In the above we have assumed that $S$ is such that any line parallel to the $z$ axis intersects $S$ in only one point. In case $S$ is not of this type, we can usually subdivide $S$ into surfaces $S_1,\ S_2,\ \dots$ which are of this type. Then the surface integral over $S$ is defined as the sum of the surface integrals over $S_1,\ S_2,\ \dots$

The results stated hold when $S$ is projected on to a region $\cal R$ of the $xy$ plane. In some cases it is better to project $S$ on to the $yz$ or $xz$ planes. For such cases (30) can be evaluated by appropriately modifying (33) and (34).