Let be an open, two-sided surface bounded by a closed non-intersecting curve (simple closed curve). Consider a directed line normal to as positive if it is on one side of , and negative if it is on the other side of . The choice of which side is positive is arbitrary but should be decided upon in advance. Call the direction or sense of positive if an observer, walking on the boundary of with his head pointing in the direction of the positive normal, has the surface on his left. Then if are single-valued, continuous, and have continuous first partial derivatives in a region of space including , we have
In vector form with and , this is simply expressed as
In words this theorem, called Stoke’s theorem, states that the line integral of the tangential component of a vector taken around a simple closed curve is equal to the surface integral of the normal component of the curl of taken over any surface having as a boundary. Note that if, as a special case in (39), we obtain the result (28).
Let be a closed surface bounding a region of volume . Choose the outward drawn normal to the surface as the positive normal and assume that are the angles which this normal makes with the positive , and axes respectively. Then if and are continuous and have continuous partial derivatives in the region
which can also be written
In vector form with and , these can be simply written as
In words this theorem, called the divergence theorem or Green’s theorem in space, states that the surface is equal to the integral of the normal component of a vector taken over a closed surface is equal to the integral of the divergence of taken over the volume enclosed by the surface.
ガウスの発散定理とも呼ばれる定理です．発散を，ベクトル場 内における容積 V の単位体積あたりの湧出量と捉えると，定理の左辺の意味は『容積 V 内全体での流量の変化量』を表わすと考えられ，右辺は『この容積 V の表面 S における の法線方向成分』と考えられます．流量には水流，電場，磁場などを考えます．
Let be a two-sided surface having projection on the plane as in the adjoining Fig. 6-3. Assume that an equation for is , where is single-valued and continuous for all and in . Divide into subregions of area , and erect a vertical column on each of these subregions to intersect in an area .
Let be single-valued and continuous at all points of . Form the sum
where is some point of . If the limit of this sum as in such a way that each exists, the resulting limit is called the surface integral of over and is designated by
Since approximately, where is the angle between the normal line to and the positive axis, the limit of the sum (29) can be written
The quantity is given by
Then assuming that has continuous (or sectionally continuous) derivatives in , (31) can be written in rectangular form as
In case the equation for is given as , (33) can also be written
The results (33) or (34) can be used to evaluate (30).
In the above we have assumed that is such that any line parallel to the axis intersects in only one point. In case is not of this type, we can usually subdivide into surfaces which are of this type. Then the surface integral over is defined as the sum of the surface integrals over
The results stated hold when is projected on to a region of the plane. In some cases it is better to project on to the or planes. For such cases (30) can be evaluated by appropriately modifying (33) and (34).
A necessary and sufficient condition for to be independent of the path joining any two given points in a region is that in
where it is supposed that these partial derivatives are continuous in .
The condition (23) is also the condition that is an exact differential, i.e. that there exists a function such that . In such case if the end points of curve are and , the value of the line integral is given by
In particular if (23) holds and is closed, we have and
The results in Theorem 6-1 can be extended to line integrals in space. Thus we have
Theorem 6-2.
A necessary and sufficient condition for to be independent of the path joining any two given points in a region is that in
where it is supposed that these partial derivatives are continuous in .
The results can be expressed concisely in terms of vectors. If , the line integral can be written and condition (26) is equivalent to the condition . If represents a force field which acts on an object, the result is equivalent to the statement that the work done in moving the object from one point to another is independent of the path joining the two points if and only if . Such a force field is often called conservative.
The condition (26) [or the equivalent condition ] is also the condition that [or ] is an exact differential, i.e. that there exists a function such that . In such case if the endpoints of curve are and , the value of the line integral is given by
A simple closed curve is a curve which does not intersect itself anywhere. Mathematically, a curve in the plane is defined by the parametric equations where and are single-valued and continuous in an interval . If and , the curve is said to be closed. If and only when (except in the special case where and ), the curve is closed and does not intersect itself and so is a simple closed curve. We shall also assume, unless otherwise stated, that and are piecewise differentiable in .
If a plane region has the property that any closed curve in it can be continuously shrunk to a point without leaving the region, then the region is called simply-connected, otherwise it is called multiply-connected.
As the parameter varies from to , the plane curve is described in a certain sense or direction. For curves in the plane, we arbitrarily describe this direction as positive or negative according as a person traversing the curve in this direction with his head pointing in the positive direction has the region enclosed by the curve always toward his left or right respectively. If we look down upon a simple closed curve in the plane, this amounts to saying that traversal of the curve in the counterclockwise direction is taken as positive while traversal in the clockwise direction is taken as negative.
If the equation of a curve C in the plane is given as , the line integral (14) is evaluated by placing in the integrand to obtain the definite integral
which is then evaluated in the usual manner.
Similarly if C is given as , then and the line integral becomes
If C is given in parametric form , the line integral becomes
where and denote the values of corresponding to points and respectively.
Combination of the above methods may be used in the evaluation.
Similar methods are used for evaluating line integrals along space curve.
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
where and . The line integral (14) is a special case of this with .
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
represents physically the total work done in moving the object along the curve C.
Let C be a curve in the xy plane which connects points and , (see Fig. 6-2). Let and be single-valued functions defined at all points of C. Subdivide C into n parts by choosing n – 1 points on it given by . Call and and suppose that points are chosen so that they are situated on C between points and . Form the sum
The limit of this sum as in such a way that all quantities approaches zero, if such limit exists, is called a line integral along C and is denoted by
or
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 and .
In an exactly analogous manner one may define a line integral along a curve C in three dimensional space as
where , and are functions of , and .
Other types of line integrals, depending on particular curves, can be defined. For example, if denotes the arc length along curve C in the above figure between points and , then
is called the line integral of along curve C. Extensions to three (or higher) dimensions are possible.
be an open, two-sided surface bounded by a closed non-intersecting curve
(simple closed curve). Consider a directed line normal to
are single-valued, continuous, and have continuous first partial derivatives in a region of space including
![\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)](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cint_C%5BA_1dx+%2B+A_2dy+%2B+A_3dz%5D+%3D%5C%5C%5Cvspace%7B0.2in%7D+%5Cunderset%7BS%7D%7B%5Ciint%7D%5Cleft%5B+%5Cleft%28+%5Cfrac%7B%5Cpartial+A_3%7D%7B%5Cpartial+y%7D+-%5Cfrac%7B%5Cpartial+A_2%7D%7B%5Cpartial+z%7D+%5Cright%29%5Ccos%5Calpha+%2B+%5Cleft%28+%5Cfrac%7B%5Cpartial+A_1%7D%7B%5Cpartial+z%7D+-%5Cfrac%7B%5Cpartial+A_3%7D%7B%5Cpartial+x%7D+%5Cright%29%5Ccos%5Cbeta+%2B+%5Cleft%28+%5Cfrac%7B%5Cpartial+A_2%7D%7B%5Cpartial+x%7D+-%5Cfrac%7B%5Cpartial+A_1%7D%7B%5Cpartial+y%7D+%5Cright%29%5Ccos%5Cgamma+%5Cright%5DdS+%5Ccdots%2838%29&bg=T&fg=000000&s=0 "\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)")
and
, this is simply expressed as
taken around a simple closed curve
taken over any surface
in (39), we obtain the result (28).
. Choose the outward drawn normal to the surface as the positive normal and assume that
are the angles which this normal makes with the positive
,
and
axes respectively. Then if
and
are continuous and have continuous partial derivatives in the region
![\displaystyle \underset{V}{\iiint}\left( \frac{\partial A_1}{\partial x} + \frac{\partial A_2}{\partial y} + \frac{\partial A_3}{\partial z} \right)dV = \underset{S}{\iint}[ A_1dydz + A_2dzdx + A_3dxdy ]\cdots (36)](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cunderset%7BV%7D%7B%5Ciiint%7D%5Cleft%28+%5Cfrac%7B%5Cpartial+A_1%7D%7B%5Cpartial+x%7D+%2B+%5Cfrac%7B%5Cpartial+A_2%7D%7B%5Cpartial+y%7D+%2B+%5Cfrac%7B%5Cpartial+A_3%7D%7B%5Cpartial+z%7D+%5Cright%29dV+%3D+%5Cunderset%7BS%7D%7B%5Ciint%7D%5B+A_1dydz+%2B+A_2dzdx+%2B+A_3dxdy+%5D%5Ccdots+%2836%29&bg=T&fg=000000&s=0 "\displaystyle \underset{V}{\iiint}\left( \frac{\partial A_1}{\partial x} + \frac{\partial A_2}{\partial y} + \frac{\partial A_3}{\partial z} \right)dV = \underset{S}{\iint}[ A_1dydz + A_2dzdx + A_3dxdy ]\cdots (36)")
and
, these can be simply written as
on the
plane as in the adjoining Fig. 6-3. Assume that an equation for
, where
is single-valued and continuous for all
subregions of area
, and erect a vertical column on each of these subregions to intersect
.
be single-valued and continuous at all points of
is some point of
in such a way that each
exists, the resulting limit is called the surface integral of
over
approximately, where
is the angle between the normal line to
is given by
has continuous (or sectionally continuous) derivatives in
, (33) can also be written
which are of this type. Then the surface integral over
or
planes. For such cases (30) can be evaluated by appropriately modifying (33) and (34).
to be independent of the path
is an exact differential, i.e. that there exists a function
such that
. In such case if the end points of curve
and
, the value of the line integral is given by
![\displaystyle \int_{(x_1, y_1)}^{(x_2, y_2)}[Pdx + Qdy] = \int_{(x_1, y_1)}^{(x_2, y_2)} d\phi = \phi(x_2, y_2) - \phi(x_1, y_1) \cdots(24)](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cint_%7B%28x_1%2C+y_1%29%7D%5E%7B%28x_2%2C+y_2%29%7D%5BPdx+%2B+Qdy%5D+%3D+%5Cint_%7B%28x_1%2C+y_1%29%7D%5E%7B%28x_2%2C+y_2%29%7D+d%5Cphi+%3D+%5Cphi%28x_2%2C+y_2%29+-+%5Cphi%28x_1%2C+y_1%29+%5Ccdots%2824%29&bg=T&fg=000000&s=0 "\displaystyle \int_{(x_1, y_1)}^{(x_2, y_2)}[Pdx + Qdy] = \int_{(x_1, y_1)}^{(x_2, y_2)} d\phi = \phi(x_2, y_2) - \phi(x_1, y_1) \cdots(24)")
and
![\displaystyle \oint_C [Pdx + Qdy] = 0\cdots(25)](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Coint_C+%5BPdx+%2B+Qdy%5D+%3D+0%5Ccdots%2825%29&bg=T&fg=000000&s=0 "\displaystyle \oint_C [Pdx + Qdy] = 0\cdots(25)")
to be independent of the path
and condition (26) is equivalent to the condition
. If
which acts on an object, the result is equivalent to the statement that the work done in moving the object from one point to another is independent of the path joining the two points if and only if
] is also the condition that
[or
] is an exact differential, i.e. that there exists a function
. In such case if the endpoints of curve
and
, the value of the line integral is given by
be single-valued and continuous in a simply-connected region
![\displaystyle \oint_C [Pdx + Qdy] = \underset{\cal R}{\iint}\left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right)dxdy\cdots(22)](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Coint_C+%5BPdx+%2B+Qdy%5D+%3D+%5Cunderset%7B%5Ccal+R%7D%7B%5Ciint%7D%5Cleft%28+%5Cfrac%7B%5Cpartial+Q%7D%7B%5Cpartial+x%7D+-+%5Cfrac%7B%5Cpartial+P%7D%7B%5Cpartial+y%7D+%5Cright%29dxdy%5Ccdots%2822%29&bg=T&fg=000000&s=0 "\displaystyle \oint_C [Pdx + Qdy] = \underset{\cal R}{\iint}\left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right)dxdy\cdots(22)")
is used to emphasize that
where
and
are single-valued and continuous in an interval
. If
and
, the curve is said to be closed. If
and
only when
(except in the special case where
and
), the curve is closed and does not intersect itself and so is a simple closed curve. We shall also assume, unless otherwise stated, that
and
are piecewise differentiable in
varies from
to
, the plane curve is described in a certain sense or direction. For curves in the
is another point on C.
![\displaystyle \int_C [P(x, y)dx + Q(x, y)dy] = \int_C P(x, y)dx + \int_C Q(x, y)dy](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cint_C+%5BP%28x%2C+y%29dx+%2B+Q%28x%2C+y%29dy%5D+%3D+%5Cint_C+P%28x%2C+y%29dx+%2B+%5Cint_C+Q%28x%2C+y%29dy&bg=T&fg=000000&s=0 "\displaystyle \int_C [P(x, y)dx + Q(x, y)dy] = \int_C P(x, y)dx + \int_C Q(x, y)dy")
![\displaystyle \int_{(a_1, b_1)}^{(a_2, b_2)} [Pdx + Qdy] = - \int_{(a_2, b_2)}^{(a_1, b_1)}[Pdx + Qdy]](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cint_%7B%28a_1%2C+b_1%29%7D%5E%7B%28a_2%2C+b_2%29%7D+%5BPdx+%2B+Qdy%5D+%3D+-+%5Cint_%7B%28a_2%2C+b_2%29%7D%5E%7B%28a_1%2C+b_1%29%7D%5BPdx+%2B+Qdy%5D&bg=T&fg=000000&s=0 "\displaystyle \int_{(a_1, b_1)}^{(a_2, b_2)} [Pdx + Qdy] = - \int_{(a_2, b_2)}^{(a_1, b_1)}[Pdx + Qdy]")
![\displaystyle \int_{(a_1, b_1)}^{(a_2, b_2)} [Pdx + Qdy] = \int_{(a_1, b_1)}^{(a_3, b_3)} [Pdx + Qdy] + \int_{(a_3, b_3)}^{(a_2, b_2)} [Pdx + Qdy]](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cint_%7B%28a_1%2C+b_1%29%7D%5E%7B%28a_2%2C+b_2%29%7D+%5BPdx+%2B+Qdy%5D+%3D+%5Cint_%7B%28a_1%2C+b_1%29%7D%5E%7B%28a_3%2C+b_3%29%7D+%5BPdx+%2B+Qdy%5D+%2B+%5Cint_%7B%28a_3%2C+b_3%29%7D%5E%7B%28a_2%2C+b_2%29%7D+%5BPdx+%2B+Qdy%5D+&bg=T&fg=000000&s=0 "\displaystyle \int_{(a_1, b_1)}^{(a_2, b_2)} [Pdx + Qdy] = \int_{(a_1, b_1)}^{(a_3, b_3)} [Pdx + Qdy] + \int_{(a_3, b_3)}^{(a_2, b_2)} [Pdx + Qdy]")
is given as
, the line integral (14) is evaluated by placing
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)](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cint_%7Ba_1%7D%5E%7Ba_2%7D%5BP%5C%7Bx%2C+f%28x%29%5C%7Ddx+%2B+Q%5C%7Bx%2C+f%28x%29%5C%7Df%27%28x%29dx%5D+%5Ccdots%2819%29&bg=T&fg=000000&s=0 "\displaystyle \int_{a_1}^{a_2}[P\{x, f(x)\}dx + Q\{x, f(x)\}f'(x)dx] \cdots(19)")
, then
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)](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cint_%7Bb_1%7D%5E%7Bb_2%7D%5BP%5C%7Bg%28y%29%2C+y%5C%7Dg%27%28y%29dy+%2B+Q%5C%7Bg%28y%29%2C+y%5C%7Ddy%5D%5Ccdots%2820%29&bg=T&fg=000000&s=0 "\displaystyle \int_{b_1}^{b_2}[P\{g(y), y\}g'(y)dy + Q\{g(y), y\}dy]\cdots(20)")
![\displaystyle \int_{t_1}^{t_2} [P\{ \phi(t),\ \psi(t) \}\phi'(t)dt + Q\{ \phi(t),\ \psi(t) \}\psi'(t)dt] \cdots (21)](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cint_%7Bt_1%7D%5E%7Bt_2%7D+%5BP%5C%7B+%5Cphi%28t%29%2C%5C+%5Cpsi%28t%29+%5C%7D%5Cphi%27%28t%29dt+%2B+Q%5C%7B+%5Cphi%28t%29%2C%5C+%5Cpsi%28t%29+%5C%7D%5Cpsi%27%28t%29dt%5D+%5Ccdots+%2821%29&bg=T&fg=000000&s=0 "\displaystyle \int_{t_1}^{t_2} [P\{ \phi(t),\ \psi(t) \}\phi'(t)dt + Q\{ \phi(t),\ \psi(t) \}\psi'(t)dt] \cdots (21)")
and
respectively.
![\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)](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cint_%7BC%7D%5BA_1dx+%2B+A_2dy+%2B+A_3dz%5D+%5C%5C++%3D+%5Cint_%7BC%7D+%28A_1%5Cbold%7Bi%7D+%2B+A_2%5Cbold%7Bj%7D+%2B+A_3%5Cbold%7Bk%7D%29+%5Ccdot+%28dx%5Cbold%7Bi%7D+%2B+dy%5Cbold%7Bj%7D+%2B+dz%5Cbold%7Bk%7D%29%5C%5C++%3D+%5Cint_%7BC%7D+%5Cbold%7BA%7D%5Ccdot+d%5Cbold%7Br%7D+%5Ccdots+%2817%29+&bg=T&fg=000000&s=0 "\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)")
. The line integral (14) is a special case of this with
.
and
, (see Fig. 6-2). Let
and
be single-valued functions defined at all points of C. Subdivide C into n parts by choosing n – 1 points on it given by
. Call
and
and suppose that points
are chosen so that they are situated on C between points
and
. Form the sum
in such a way that all quantities
approaches zero, if such limit exists, is called a line integral along C and is denoted by
or
and
.
![\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)](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Clim%5Climits_%7Bn+%5Crightarrow%5Cinfty%7D%5Csum_%7Bk%3D1%7D%5E%7Bn%7D%5Cleft%5C%7B+A_1%28%5Cxi_k%2C+%5Ceta_k%2C+%5Czeta_k%29%5CDelta+x_k+%2B+A_2%28%5Cxi_k%2C+%5Ceta_k%2C+%5Czeta_k%29%5CDelta+y_k+%2B+A_3%28%5Cxi_k%2C+%5Ceta_k%2C+%5Czeta_k%29%5CDelta+z_k++%5Cright%5C%7D+%5C%5C+%3D+%5Cint_C+%5Cleft%5B+A_1dx+%2B+A_2dy+%2B+A_3dz+%5Cright%5D+%5Ccdots%2815%29&bg=T&fg=000000&s=0 "\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)")
,
and
denotes the arc length along curve C in the above figure between points
, then
along curve C. Extensions to three (or higher) dimensions are possible.
![\displaystyle \int_{(a_1, b_1)}^{(a_2, b_2)}\left[ Pdx + Qdy \right]\cdots(14)](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cint_%7B%28a_1%2C+b_1%29%7D%5E%7B%28a_2%2C+b_2%29%7D%5Cleft%5B+Pdx+%2B+Qdy+%5Cright%5D%5Ccdots%2814%29&bg=T&fg=000000&s=0 "\displaystyle \int_{(a_1, b_1)}^{(a_2, b_2)}\left[ Pdx + Qdy \right]\cdots(14)")