РЯДЫ. КРАТНЫЕ ИНТЕГРАЛЫ. ТЕОРИЯ ПОЛЯ / t13

\section*{‹ҐЄжЁп 13.}

”®а¬г«  ѓ гбб -Ћбва®Ја ¤бЄ®Ј®. „ЁўҐаЈҐ­жЁп ўҐЄв®а­®Ј® Ї®«п, ҐҐ
дЁ§ЁзҐбЄЁ© б¬лб«. ”®а¬г«  ‘в®Єб . ђ®в®а ўҐЄв®а­®Ј® Ї®«п, ҐЈ®
дЁ§ЁзҐбЄЁ© б¬лб«.

ЋЎа вЁ¬бп Є а бᬮв७Ёо д®а¬г«л ѓ гбб -Ћбва®Ја ¤бЄ®Ј®. Џгбвм
$$Z(\varphi_1,\varphi_2)=\{(x,y,z)\in\mathbb{R}^3\ |\ (x,y)\in T,\
\varphi_1(x,y) \leqslant z \leqslant\varphi_2(x,y)\},\qquad
T\in\mathcal{T}_2,$$ $\varphi_1(\cdot)$, $\varphi_2(\cdot)$ --
­ҐЇаҐалў­лҐ ­  $T$ дг­ЄжЁЁ, ЇаЁ­ ¤«Ґ¦ йЁҐ Є« ббг
$C^1(\widehat{T})$, $\varphi_1(x,y)\leqslant\varphi_2(x,y)$ ЇаЁ
Є ¦¤®¬ $(x,y)\in T$. ЋЄ §лў Ґвбп, зв® Ја ­Ёжг $\Gamma$ ¬­®¦Ґбвў 
$Z$, б®бв®пйго Ё§ $\Gamma_0$ -- Ў®Є®ў®© Ї®ўҐае­®бвЁ $Z$ Ё
Ї®ўҐае­®б⥩ $$\Gamma_1=\{(x,y,\varphi_1(x,y))\in\mathbb{R}^3\ |\
(x,y)\in T \},\quad
\Gamma_2=\{(x,y,\varphi_2(x,y))\in\mathbb{R}^3\ |\ (x,y)\in T\}$$
¬®¦­® ®аЁҐ­вЁа®ў вм. ќвг ®аЁҐ­в жЁо, в.Ґ. дг­ЄжЁо $n(M)$,
$M\in\Gamma$, ¬®¦­® ўлЎа вм в Є, зв®Ўл Ї®б«Ґ¤­пп Є®¬Ї®­Ґ­в 
ўҐЄв®а  $n(M)$ ў® ў­гв७­Ёе в®зЄ е $M\in\Gamma_2$, Ўл«  Ўл Ў®«миҐ
­г«п (Ё­л¬Ё б«®ў ¬Ё ўҐЄв®а $n(M)$ б®бв ў«п« Ўл ®бвал© гЈ®« б
­ Їа ў«Ґ­ЁҐ¬ ®бЁ $Oz$),   ў® ў­гв७­Ёе в®зЄ е $M\in\Gamma_1$,
Ўл«  Ўл ¬Ґ­миҐ ­г«п (Ё­л¬Ё б«®ў ¬Ё ўҐЄв®а $n(M)$ б®бв ў«п« Ўл
вгЇ®© гЈ®« б ­ Їа ў«Ґ­ЁҐ¬ ®бЁ $Oz$). ЏаЁ н⮬ ®аЁҐ­в жЁп, в.Ґ.
§­ зҐ­Ёп $n(M)$ ў в®зЄ е Ї®ўҐае­®бвЁ $\Gamma_0$ (нв  з бвм
Ї®ўҐае­®бвЁ б®бв®Ёв Ё§ ®в१Є®ў Їап¬ле, Ї а ««Ґ«м­ле ®бЁ $Oz$,
Ўлвм ¬®¦Ґв ўл஦¤ ойЁебп ў в®зЄг) ¤«п ¤ «м­Ґ©иҐЈ® ­Ґ в Є Ё ў ¦­ .
‚ ¦­® «Ёим в®, зв® нвЁ §­ зҐ­Ёп (®­Ё ЇаҐ¤бв ў«пов Ё§ бҐЎп ўҐЄв®а 
ў $\mathbb{R}^3$) ЇҐаЇҐ­¤ЁЄг«па­л ®бЁ $Oz$. ‚лЎа ­­го ®аЁҐ­в жЁо
Ї®ўҐае­®бвЁ $\Gamma$ Ўг¤Ґ¬ ®в¬Ґз вм §­ Є®¬ $"+"$: $\Gamma_0^+$,
$\Gamma_1^+$, $\Gamma_2^+$. „«п ЇаЁ¬Ґа  ўлзЁб«Ё¬ Є®¬Ї®­Ґ­вл
­®а¬ «Ё ¤«п $\Gamma_2^+$: $${\rm det}\begin{vmatrix}i& j& k\\1& 0&
\frac{\partial\varphi_2}{\partial x}\\0& 1&
\frac{\partial\varphi_2}{\partial y}\end{vmatrix}=
\bigl(-\frac{\partial\varphi_2}{\partial
x},-\frac{\partial\varphi_2}{\partial y},1\bigr)$$ Ё, §­ зЁв,
ваҐвмп Є®¬Ї®­Ґ­в  ўҐЄв®а  $n(M)$ ¤«п $M\in\Gamma_2$ а ў­ 
$(1+(\frac{\partial\varphi_2}{\partial x})^2
+(\frac{\partial\varphi_2}{\partial y})^2)^{-1/2}$, ўлзЁб«Ґ­­®¬г ў
в®зЄҐ $M$. Ђ­ «®ЈЁз­®, ваҐвмп Є®¬Ї®­Ґ­в  нв®Ј® ўҐЄв®а  ¤«п
$M\in\Gamma_1$ а ў­  $-(1+(\frac{\partial\varphi_1}{\partial x})^2
+(\frac{\partial\varphi_1}{\partial y})^2)^{-1/2}$.

Џгбвм ⥯Ґам $f(\cdot)$ -- ­ҐЇаҐалў­ п ­  ®Ў« бвЁ $Z$ дг­ЄжЁп,
®Ў« ¤ ой п ­ҐЇаҐалў­®© ­  $Z$ з бв­®© Їа®Ё§ў®¤­®© $\frac{\partial
f}{\partial z}$. ’®Ј¤  $$\int\!\!\int\!\!\int_{Z}\frac{\partial
f}{\partial z}\,dx\,dy\,dz=\int\!\!\!\int_T
\left(\int_{\varphi_1(x,y)}^{\varphi_2(x,y)} \frac{\partial
f}{\partial z}\,dz\right)\,dx\,dy=\int\!\!\!\int_T \bigl(
f(x,y,\varphi_2(x,y))-f(x,y,\varphi_1(x,y))\bigr)\,dx\,dy=$$
$$=\int\!\!\!\int_T f(x,y,\varphi_2(x,y))
\left(1+(\frac{\partial\varphi_2}{\partial x})^2
+(\frac{\partial\varphi_2}{\partial y})^2\right)^{-1/2}
\left(1+(\frac{\partial\varphi_2}{\partial x})^2
+(\frac{\partial\varphi_2}{\partial y})^2\right)^{1/2}\,dx\,dy-$$
$$-\int\!\!\!\int_T f(x,y,\varphi_1(x,y))
\left(1+(\frac{\partial\varphi_1}{\partial x})^2
+(\frac{\partial\varphi_1}{\partial y})^2\right)^{-1/2}
\left(1+(\frac{\partial\varphi_1}{\partial x})^2
+(\frac{\partial\varphi_1}{\partial y})^2\right)^{1/2}\,dx\,dy=$$
$$=\int\!\!\!\int_{\Gamma_2} f(x,y,z)
\bigl(n(x,y,z),(0,0,1)\bigr)\,dS+\int\!\!\!\int_{\Gamma_1}
f(x,y,z) \bigl(n(x,y,z),(0,0,1)\bigr)\,dS.$$ ’ Є Є Є
$\bigl(n(x,y,z),(0,0,1)\bigr)=0$ ¤«п $(x,y,z)\in\Gamma_0$, в®
®Є®­з вҐ«м­ п д®а¬г«  Ё¬ҐҐв ўЁ¤
$$\int\!\!\int\!\!\int_{Z}\frac{\partial f}{\partial
z}\,dx\,dy\,dz=\int\!\!\!\int_{\Gamma^+} f(x,y,z)\,dx\,dy$$ --
ва®©­®© Ё­вҐЈа « Ї® ¬­®¦Ґбвўг $Z$ ў ваҐе¬Ґа­®¬ Їа®бва ­б⢥
бў®¤Ёвбп Є Ї®ўҐае­®бв­®¬г Ё­вҐЈа «г ўв®а®Ј® த  Ї® Ја ­ЁжҐ нв®Ј®
¬­®¦Ґбвў  (б ­ ¤«Ґ¦ йЁ¬ ®Ўа §®¬ ўлЎа ­­®© ®аЁҐ­в жЁҐ© нв®©
Ја ­Ёжл). ќв  д®а¬г«  ®бв Ґвбп бЇа ўҐ¤«Ёў®© Ё ¤«п Ў®«ҐҐ б«®¦­®
гбв஥­­ле ®Ў« б⥩ $W\in\mathcal{T}_3$. €¬Ґ­­®, Їгбвм ®Ў« бвм $W$
¬®¦Ґв Ўлвм а §¤Ґ«Ґ­  ­  Є®­Ґз­®Ґ зЁб«® з б⥩ $\{Z_i\}_{i=1}^n$
а бᬮв७­®Ј® ўЁ¤  (Є Є ®Ўлз­® ЇаҐ¤Ї®« Ј Ґвбп, зв® ­ЁЄ ЄЁҐ ¤ўҐ Ё§
нвЁе з б⥩ ­Ґ Ё¬Ґов ®ЎйЁе ў­гв७­Ёе в®зҐЄ). ’®Ј¤  ¬®¦­®
Ї®Є § вм, зв® Ја ­Ёж  $\Gamma$ ®Ў« бвЁ $W$ пў«пҐвбп ®аЁҐ­вЁа㥬®©
Ї®ўҐае­®бвмо. ‚лЎЁа п Ї®«®¦ЁвҐ«м­®© ®аЁҐ­в жЁҐ© $\Gamma^+$ вг, ЇаЁ
Є®в®а®© ­ Їа ў«Ґ­ЁҐ ­®а¬ «Ё Є Ї®ўҐае­®бвЁ $\Gamma$ пў«пҐвбп
ў­Ґи­Ё¬ Ї® ®в­®иҐ­Ёо Є ®Ў« бвЁ $W$, ¬®¦­® гбв ­®ўЁвм, зв®
$$\int\!\!\int\!\!\int_{W}\frac{\partial f}{\partial
z}\,dx\,dy\,dz=\int\!\!\!\int_{\Gamma^+} f(x,y,z)\,dx\,dy$$ (ў
ЇаҐ¤Ї®«®¦Ґ­ЁЁ ­ҐЇаҐалў­®бвЁ дг­ЄжЁЁ $f$ Ё ҐҐ з бв­®© Їа®Ё§ў®¤­®©
Ї® ЇҐаҐ¬Ґ­­®© $z$ ­  ®Ў« бвЁ $W$). ќвг д®а¬г«г Ё ¬®¦­® Ўл«® Ўл
­ §ў вм д®а¬г«®© ѓ гбб -Ћбва®Ја ¤бЄ®Ј®, ®¤­ Є® ®Ўлз­® ҐҐ
§ ЇЁблў ов ў Ў®«ҐҐ бЁ¬¬ҐваЁз­®¬ ўЁ¤Ґ. ЏаЁ н⮬ ¤ ў ©вҐ ᤥ« Ґ¬
¤®Ї®«­ЁвҐ«м­®Ґ ЇаҐ¤Ї®«®¦Ґ­ЁҐ ® ў®§¬®¦­®бвЁ а §¤Ґ«Ёвм ®Ў« бвм $W$
­  Є®­Ґз­®Ґ зЁб«® з б⥩ $\{Z_i\}_{i=1}^n$ ­Ґ в®«мЄ® ў ­ Їа ў«Ґ­ЁЁ
®бЁ $Oz$, ­® в Є¦Ґ Ё ў ­ Їа ў«Ґ­ЁЁ Є ¦¤®© Ё§ ¤ўге ¤агЈЁе: $Ox$ Ё
$Oy$. Џгбвм $f$, $g$, $h$ - ­ҐЇаҐалў­лҐ ­  $W$ дг­ЄжЁЁ Є« бб 
$C^1(W)$. ’®Ј¤  $$\int\!\!\int\!\!\int_{W}\left(\frac{\partial
h}{\partial x}+\frac{\partial g}{\partial y}+\frac{\partial
f}{\partial z}\right)\,dx\,dy\,dz=\int\!\!\!\int_{\Gamma^+}
h(x,y,z)\,dy\,dz+g(x,y,z)\,dz\,dx+f(x,y,z)\,dx\,dy.$$ ќв  д®а¬г« 
­ §лў Ґвбп д®а¬г«®© ѓ гбб -Ћбва®Ја ¤бЄ®Ј®. —Ёб«® $${\rm div}\, a=
\frac{\partial h}{\partial x}+\frac{\partial g}{\partial
y}+\frac{\partial f}{\partial z}$$ ­ §лў Ґвбп ¤ЁўҐаЈҐ­жЁҐ©
ўҐЄв®а­®Ј® Ї®«п $a=(h,g,f)$. ЏаЁ¬Ґ­пп ᤥ« ­­лҐ ®ЇаҐ¤Ґ«Ґ­Ёп,
д®а¬г«г ѓ гбб -Ћбва®Ја ¤бЄ®Ј® ¬®¦­® § ЇЁб вм Є Є
$$\int\!\!\int\!\!\int_{W}{\rm div}\,
a\,dx\,dy\,dz=\int\!\!\!\int_{\Gamma} \bigl(a,n\bigr)\,dS\eqno
(1)$$ -- ва®©­®© Ё­вҐЈа « Ї® ¬­®¦Ґбвўг $W$ ®в ¤ЁўҐаЈҐ­жЁЁ
ўҐЄв®а­®Ј® Ї®«п $a=(h,g,f)$ бў®¤Ёвбп Є Ї®ўҐае­®бв­®¬г Ё­вҐЈа «г
ЇҐаў®Ј® த  Ї® Ја ­ЁжҐ $\Gamma$ нв®Ј® ¬­®¦Ґбвў  ®в бЄ «па­®Ј®
Їа®Ё§ўҐ¤Ґ­Ёп $\bigl(a,n\bigr)$ нв®Ј® ўҐЄв®а­®Ј® Ї®«п Ё ў­Ґи­Ґ©, Ї®
®в­®иҐ­Ёо Є $W$, ­®а¬ «Ё Є $\Gamma$ (Ё­ зҐ, -- а ўҐ­ Ї®в®Єг нв®Ј®
ўҐЄв®а­®Ј® Ї®«п ў ­ Їа ў«Ґ­ЁЁ ў­Ґи­Ґ© ­®а¬ «Ё Є $\Gamma$). …б«Ё
ў®бЇаЁ­Ё¬ вм Ї®«Ґ $a$ Є Є бЄ®а®бвм ¤ўЁ¦г饩бп ¦Ё¤Є®бвЁ, в®
Ё­вҐЈа « ў Їа ў®© з бвЁ (1) ЇаҐ¤бв ў«пҐв Ё§ бҐЎп Є®«ЁзҐбвў®
¦Ё¤Є®бвЁ, Їа®вҐЄ о饩 зҐаҐ§ Ї®ўҐае­®бвм $\Gamma$ ў § ¤ ­­го
бв®а®­г ў Ґ¤Ё­Ёж㠢६Ґ­Ё. —в®Ўл нв® Є®«ЁзҐбвў® (Ї®в®Є) Ўл«®
®в«Ёз­® ®в ­г«п, б®Ј« б­® (1), ­Ґ®Ўе®¤Ё¬®, зв®Ўл ў­гваЁ $W$
­ е®¤Ё«Ёбм Ёбв®з­ЁЄЁ (Ё«Ё бв®ЄЁ) ¦Ё¤Є®бвЁ (в.Ґ. ${\rm div}\,a\ne
0$). ’ ЄЁ¬ ®Ўа §®¬, ¤ЁўҐаЈҐ­жЁп е а ЄвҐаЁ§гҐв Ёбв®з­ЁЄЁ Ї®«п $a$.

ЋЎбг¤Ё¬ ⥯Ґам ҐйҐ ®¤­г д®а¬г«г, ®Ў®Ўй ойго 㦥 а бᬮв७­го
д®а¬г«г ѓаЁ­ . Ќ Ї®¬­Ё¬, зв® д®а¬г«  ѓаЁ­  Ї®§ў®«пҐв ᢥбвЁ
ўлзЁб«Ґ­ЁҐ ¤ў®©­®Ј® Ё­вҐЈа «  Ї® Ї«®бЄ®© ®Ў« бвЁ $S$ Є ўлзЁб«Ґ­Ёо
ЄаЁў®«Ё­Ґ©­®Ј® Ё­вҐЈа «  Ї® Ја ­ЁжҐ нв®© ®Ў« бвЁ. ЏаЁ ­ҐЄ®в®але
ЇаҐ¤Ї®«®¦Ґ­Ёпе ®в­®бЁвҐ«м­® $S$, $f$ Ё $g$ ®­  ўлЈ«п¤Ёв в Є
$$\int\!\!\int_S\left(\frac{\partial f}{\partial x}
-\frac{\partial g} {\partial y}\right)\,dx\,dy=\int_{\partial
S^+}g(x,y)\,dx+f(x,y)\,dy.$$ ќв  д®а¬г«  ¬®¦Ґв Ўлвм а бЇа®бва ­Ґ­ 
­  Ё­вҐЈа «л Ї® ¬­®¦Ґбвў ¬, а бЇ®«®¦Ґ­­л¬ ­  ®в­®бЁвҐ«м­®
Їа®Ё§ў®«м­ле Ї®ўҐае­®бвпе, в.Ґ. ­  Ї®ўҐае­®бв­лҐ Ё­вҐЈа «л
(Є®®а¤Ё­ в­лҐ Ї«®бЄ®бвЁ -- нв® Їа®б⥩訩 ўЁ¤ Ї®ўҐае­®бвЁ ў
$\mathbb{R}^3$).  Џгбвм $dr=(dx,dy,dz)$, $n=(\cos\alpha,
\cos\beta, \cos\gamma)$ -- ­ Їа ў«Ґ­ЁҐ ­®а¬ «Ё Є $S$, ЇаЁ Є®в®а®¬
ўлЎа ­­®Ґ ¤ўЁ¦Ґ­ЁҐ Ї® Є®­вгаг $\partial S^+$ ЇаҐ¤бв ў«пҐвбп
Ї®«®¦ЁвҐ«м­л¬, $a=(f,g,h)$ - ­ҐЇаҐалў­®Ґ ­  $S$ ўҐЄв®а­®Ґ Ї®«Ґ
Є« бб  $C^1(T)$, $T$ -- ­ҐЄ®в®а п ®Ў« бвм, ᮤҐа¦ й п Ї®ўҐае­®бвм
$S$ (®Ў« бвм $T$ вॡгҐвбп, зв®Ўл ®ЇаҐ¤Ґ«Ёвм з бв­лҐ Їа®Ё§ў®¤­лҐ
дг­ЄжЁ© $f$, $g$ Ё $h$). ’®Ј¤  $$\int_{\partial
S^+}f\,dx+g\,dy+h\,dz= \int\!\!\int_S{\rm
det}\begin{vmatrix}\cos\alpha&\cos\beta& \cos\gamma\\
\frac{\partial}{\partial x}& \frac{\partial}{\partial y}&
\frac{\partial}{\partial z}\\f& g& h\end{vmatrix}\,dS.\eqno (2)$$
‚ҐЄв®а б Є®®а¤Ё­ в ¬Ё $$\frac{\partial h}{\partial
y}-\frac{\partial g}{\partial z},\qquad \frac{\partial f}{\partial
z}-\frac{\partial h}{\partial x},\qquad \frac{\partial g}{\partial
x}-\frac{\partial f}{\partial y}$$ ­ §лў Ґвбп а®в®а®¬ Ё«Ё ўЁе६
Ї®«п $a$ Ё ®Ў®§­ з Ґвбп ${\rm rot}\,a$. Њ­Ґ¬®­ЁзҐбЄ®Ґ Їа ўЁ«® ¤«п
§ ЇЁбЁ нв®Ј® ўҐЄв®а  б®бв®Ёв ў ўлЇЁблў ­ЁЁ ®ЇаҐ¤Ґ«ЁвҐ«п $${\rm
rot}\,a={\rm det}\begin{vmatrix}i& j& k\\ \frac{\partial}{\partial
x}& \frac{\partial}{\partial y}& \frac{\partial}{\partial z}\\f&
g& h\end{vmatrix}.$$ ђ®в®а е а ЄвҐаЁ§гҐв "§ ўЁе७­®бвм" Ї®«п ў
¤ ­­®© в®зЄҐ. ЏаЁ¬Ґ­пп ᤥ« ­­лҐ ®ЇаҐ¤Ґ«Ґ­Ёп, § ЇЁиҐ¬ (2) ў ўЁ¤Ґ
$$\int_{\partial S^+}(a,\,dr)= \int\!\!\int_S({\rm
rot}\,a,n)\,dS.\eqno (2')$$ ”®а¬г«л (2), $(2')$ гбв ­ ў«Ёў овбп
б­ з «  ­  Ї а ¬ҐваЁзҐбЄЁе Ї®ўҐае­®бвпе ба ў­Ґ­ЁҐ¬ «Ґў®© Ё Їа ў®©
з б⥩ а ўҐ­бвў  (2),   Ї®в®¬ а бЇа®бва ­повбп Ї®  ¤¤ЁвЁў­®бвЁ ­ 
Ў®«ҐҐ б«®¦­лҐ Ї®ўҐае­®бвЁ. ”Ё§ЁзҐбЄЁ© б¬лб« д®а¬г«л ‘в®Єб 
§ Є«оз Ґвбп ў ҐҐ Ё­вҐаЇаҐв жЁЁ Є Є $(2')$: жЁаЄг«пжЁп ўҐЄв®а­®Ј®
Ї®«п $a$ ў¤®«м § ¬Є­гв®Ј® Є®­вга  $\partial S$ а ў­  Ї®в®Єг а®в®а 
нв®Ј® ўҐЄв®а­®Ј® Ї®«п, зҐаҐ§ Ї®ўҐае­®бвм, ­ вп­гвго ­  нв®в
Є®­вга. —в®Ўл жЁаЄг«пжЁп Ўл«  ®в«Ёз­  ®в ­г«п ¤«п ¬ «®Ј® Є®­вга ,
®Єаг¦ о饣® ­ҐЄ®в®аго ўлЎа ­­го в®зЄг Ї®ўҐае­®бвЁ, Ї®«Ґ ¤®«¦­®
Ї®ў®а зЁў вмбп (Ё¬Ґвм § ўЁе७ЁҐ) ўЎ«Ё§Ё нв®© в®зЄЁ.