タグ: line integral

STOKE’S THEOREM

Let S be an open, two-sided surface bounded by a closed non-intersecting curve C (simple closed curve). Consider a directed line normal to S as positive if it is on one side of S, and negative if it is on the other side of S. The choice of which side is positive is arbitrary but should be decided upon in advance. Call the direction or sense of C positive if an observer, walking on the boundary of S with his head pointing in the direction of the positive normal, has the surface on his left. Then if A_1,\ A_2,\ A_3 are single-valued, continuous, and have continuous first partial derivatives in a region of space including S, we have

\displaystyle \int_C[A_1dx + A_2dy + A_3dz] =\\\vspace{0.2in} \underset{S}{\iint}\left[ \left( \frac{\partial A_3}{\partial y} -\frac{\partial A_2}{\partial z} \right)\cos\alpha + \left( \frac{\partial A_1}{\partial z} -\frac{\partial A_3}{\partial x} \right)\cos\beta + \left( \frac{\partial A_2}{\partial x} -\frac{\partial A_1}{\partial y} \right)\cos\gamma \right]dS \cdots(38)

In vector form with \bold{A} = A_1\bold{i} + A_2\bold{j} + A_3\bold{k} and \bold{n} = \cos\alpha\bold{i} + \cos\beta\bold{j} + \cos\gamma\bold{k}, this is simply expressed as

\displaystyle \int_C \bold{A}\cdot d\bold{r} = \underset{S}{\iint}(\nabla\times\bold{A})\cdot\bold{n}dS\cdots(39)

In words this theorem, called Stoke’s theorem, states that the line integral of the tangential component of a vector \bold{A} taken around a simple closed curve C is equal to the surface integral of the normal component of the curl of A taken over any surface S having C as a boundary. Note that if, as a special case \nabla\times\bold{A} = 0 in (39), we obtain the result (28).

EVALUATION OF LINE INTEGRALS

Fig. 6-2
Fig. 6-2

If the equation of a curve C in the plane z = 0 is given as y = f(x), the line integral (14) is evaluated by placing y = f(x),\ dy = f'(x)dx in the integrand to obtain the definite integral

\displaystyle \int_{a_1}^{a_2}[P\{x, f(x)\}dx + Q\{x, f(x)\}f'(x)dx] \cdots(19)

which is then evaluated in the usual manner.

Similarly if C is given as x = g(y), then dx = g'(y)dy and the line integral becomes

\displaystyle \int_{b_1}^{b_2}[P\{g(y), y\}g'(y)dy + Q\{g(y), y\}dy]\cdots(20)

If C is given in parametric form x = \phi(t),\ y = \psi(t), the line integral becomes

\displaystyle \int_{t_1}^{t_2} [P\{ \phi(t),\ \psi(t) \}\phi'(t)dt + Q\{ \phi(t),\ \psi(t) \}\psi'(t)dt] \cdots (21)

where t_1 and t_2 denote the values of t corresponding to points A and B respectively.

Combination of the above methods may be used in the evaluation.

Similar methods are used for evaluating line integrals along space curve.

VECTOR NOTATION FOR LINE INTEGRALS

It is often convenient to express a line integral in vector form as an aid in physical or geometric understanding as well as for brevity of notation. For example, we can express the line integral (15) in the form

\displaystyle \int_{C}[A_1dx + A_2dy + A_3dz] \\ = \int_{C} (A_1\bold{i} + A_2\bold{j} + A_3\bold{k}) \cdot (dx\bold{i} + dy\bold{j} + dz\bold{k})\\ = \int_{C} \bold{A}\cdot d\bold{r} \cdots (17)

where \bold{A} = A_1\bold{i} + A_2\bold{j} + A_3\bold{k} and d\bold{r} = dx\bold{i} + dy\bold{j} + dz\bold{k}. The line integral (14) is a special case of this with z = 0.

If at each point (x, y, z) we associate a force F acting on an object (i.e. if a force field is defined), then

\displaystyle \int_{C} \bold{F}\cdot d\bold{r}\cdots(18)

represents physically the total work done in moving the object along the curve C.

LINE INTEGRALS

Fig. 6-2
Fig. 6-2

Let C be a curve in the xy plane which connects points A (a_1, b_1) and B (a_2, b_2), (see Fig. 6-2). Let P(x, y) and Q(x, y) be single-valued functions defined at all points of C. Subdivide C into n parts by choosing n – 1 points on it given by (x_1, y_1),\ (x_2, y_2),\ \dots,\ (x_{n-1}, y_{n-1}). Call \Delta x_k = x_k - x_{k-1} and \Delta y_k = y_k - y_{k-1},\ k = 1,\ 2,\ \dots\ n and suppose that points (\xi_k, \eta_k) are chosen so that they are situated on C between points (x_{k-1}, y_{k-1}) and (x_k, y_k). Form the sum

\displaystyle \sum_{k=1}^{n}\{P(\xi_k, \eta_k)\Delta x_k + Q(\xi_k, \eta_k)\Delta y_k\}\cdots(13)

The limit of this sum as n\rightarrow\infty in such a way that all quantities \Delta x_k,\ \Delta y approaches zero, if such limit exists, is called a line integral along C and is denoted by

\displaystyle \int_C \left[ P(x, y)dx + Q(x, y)dy \right] or \displaystyle \int_{(a_1, b_1)}^{(a_2, b_2)}\left[ Pdx + Qdy \right]\cdots(14)

The limit does exist if P and Q are continuous (or piecewise continuous) at all points of C. The value of the integral depends in general on P, Q, the particular curve C, and on the limits (a_1, b_1) and (a_2, b_2).

In an exactly analogous manner one may define a line integral along a curve C in three dimensional space as

\displaystyle \lim\limits_{n \rightarrow\infty}\sum_{k=1}^{n}\left\{ A_1(\xi_k, \eta_k, \zeta_k)\Delta x_k + A_2(\xi_k, \eta_k, \zeta_k)\Delta y_k + A_3(\xi_k, \eta_k, \zeta_k)\Delta z_k \right\} \\ = \int_C \left[ A_1dx + A_2dy + A_3dz \right] \cdots(15)

where A_1, A_2 and A_3 are functions of x, y and z.

Other types of line integrals, depending on particular curves, can be defined. For example, if \Delta s_k denotes the arc length along curve C in the above figure between points (x_k, y_k) and (x_{k+1}, y_{k+1}), then

\displaystyle \lim\limits_{n \rightarrow \infty} \sum_{k=1}^{n} U(\xi_k, \eta_k)\Delta s_k = \int_C U(x, y)ds\cdots(16)

is called the line integral of U(x, y) along curve C. Extensions to three (or higher) dimensions are possible.