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\section*{‹ҐЄжЁп 10.}

ЉаЁў®«Ё­Ґ©­лҐ Ё­вҐЈа «л ЇҐаў®Ј® த  Ё ўв®а®Ј® த , Ёе бў®©бвў  Ё
ўлзЁб«Ґ­ЁҐ. ‘Є «па­®Ґ Ё ўҐЄв®а­®Ґ Ї®«Ґ. –ЁаЄг«пжЁп ўҐЄв®а­®Ј® Ї®«п
ў¤®«м ЄаЁў®©. ”®а¬г«  ѓаЁ­ .\\ ђ бᬮв७ЁҐ Ї®­пвЁп ЄаЁў®«Ё­Ґ©­®Ј®
Ё­вҐЈа «  ЇҐаў®Ј® த  ®Ўлз­® ­ зЁ­ ов б гЄ § ­Ёп в®Ј® Ї® Є Є®¬г
¬­®¦Ґбвўг нв®в Ё­вҐЈа « ўлзЁб«пҐвбп. ‚ Є зҐб⢥ в ЄЁе ¬­®¦Ґбвў
ўлбвгЇ ов ЄаЁўлҐ ­  Ї«®бЄ®бвЁ $\mathbb{R}^2$ Ё«Ё ў Їа®бва ­б⢥
$\mathbb{R}^3$. Џа®б⥩訬 ЇаЁ¬Ґа®¬ ЄаЁў®© ­  Ї«®бЄ®бвЁ пў«пҐвбп
®Ўа § $\gamma$ ®в१Є  $[a,b]\subset\mathbb{R}$, § ¤ ў Ґ¬л© Ї а®©
®в®Ўа ¦Ґ­Ё© $x,y\in C[a,b]$ Ї® Їа ўЁ«г: $$\gamma=\{(x(t),y(t))\ |\
t\in [a,b]\},\eqno (1)$$ в Є зв® а §­л¬ §­ зҐ­Ёп¬ Ї а ¬Ґва 
$t\in(a,b)$ ᮮ⢥вбвўгов а §­лҐ в®зЄЁ $(x(t),y(t))$, б®бв ў«пойЁҐ
ЄаЁўго $\gamma$. Ља®¬Ґ ­ҐЇаҐалў­®бвЁ дг­ЄжЁ© $x(\cdot)$,
$y(\cdot)$ Ї®вॡ㥬 ¤®Ї®«­ЁвҐ«м­®, зв®Ўл $x(\cdot)$, $y(\cdot)\in
C^1(a,b)$. …б«Ё Є®­жл $A=(x(a),y(a))$ Ё $B=(x(b),y(b))$ ЄаЁў®©
$\gamma$ б®ўЇ ¤ ов: $A=B$, в® в Єго ЄаЁўго Ўг¤Ґ¬ ­ §лў вм
§ ¬Є­гв®©. Џгбвм $f(\cdot)\in C(\gamma)$ -- ­ҐЇаҐалў­®Ґ
®в®Ўа ¦Ґ­ЁҐ, ЇаЁ­Ё¬ о饥 ¤Ґ©б⢨⥫м­лҐ §­ зҐ­Ёп ­  ¬­®¦Ґб⢥
$\gamma$. Џ®б«Ґ¤­ҐҐ гб«®ўЁҐ, ў з бв­®бвЁ, ®§­ з Ґв, зв® ў дг­ЄжЁо
¤ўге ЇҐаҐ¬Ґ­­ле $f(x,y)$ ў Є зҐб⢥  аЈг¬Ґ­в®ў $(x,y)$ ¬®¦­®
Ї®¤бв ўЁвм в®зЄЁ Ё§ ЄаЁў®© $\gamma$ Ё ®Ўа §®ў вм б«®¦­го дг­ЄжЁо
$f(x(t),y(t))$ ЇҐаҐ¬Ґ­­®© $t\in[a,b]$. Џ® ⥮६Ґ ® ­ҐЇаҐалў­®бвЁ
б«®¦­®© дг­ЄжЁЁ в Є п дг­ЄжЁп Ўг¤Ґв ­ҐЇаҐалў­®© Є Є дг­ЄжЁп
ЇҐаҐ¬Ґ­­®© $t\in[a,b]$. ЉаЁў®«Ё­Ґ©­лҐ Ё­вҐЈа «л ®в дг­ЄжЁЁ $f$ Ї®
ЄаЁў®© $\gamma$ ®ЇаҐ¤Ґ«пов ЇаЁ ¤®Ї®«­ЁвҐ«м­®¬ ЇаҐ¤Ї®«®¦Ґ­ЁЁ ®
­Ґўл஦¤Ґ­­®бвЁ: $${x'}^2(t)+{y'}^2(t)>0,\quad t\in (a,b);$$
ЄаЁў®«Ё­Ґ©­л¬ Ё­вҐЈа «®¬ ЇҐаў®Ј® த  ­ §лў ов (ў®®ЎйҐ Ј®ў®ап
­Ґб®Ўб⢥­­л© Ё­вҐЈа «) $$\int_{\gamma}f(x,y)\,ds=
\int_{[a,b]}f(x(t),y(t))\sqrt{{x'}^2(t)+{y'}^2(t)}\,dt,\eqno (2)$$
Ј¤Ґ бЁ¬ў®«®¬ $\int_{[a,b]}$ ®Ў®§­ зҐ­® Ё­вҐЈаЁа®ў ­ЁҐ Ї® $[a,b]$
®в ¬Ґ­м襣® §­ зҐ­Ёп  аЈг¬Ґ­в  Є Ў®«м襬г. Џгбвм $g(\cdot)\in
C(\gamma)$ -- ҐйҐ ®¤­  ­ҐЇаҐалў­ п ¤Ґ©б⢨⥫쭮§­ з­ п дг­ЄжЁп ­ 
$\gamma$. ‚ла ¦Ґ­ЁҐ $$\int_{{AB}}f(x,y)\,dx+g(x,y)\,dy=
\int_a^b(f(x(t),y(t))x'(t)+g(x(t),y(t))y'(t))\,dt\eqno (2')$$
­ §лў ов ЄаЁў®«Ё­Ґ©­л¬ Ё­вҐЈа «®¬ ўв®а®Ј® த  (Ё­вҐЈа « ў Їа ў®©
з бвЁ в Є¦Ґ ¬®¦Ґв Ўлвм ­Ґб®Ўб⢥­­л¬). ЏаҐ¤Ґ«л Ё­вҐЈаЁа®ў ­Ёп ў
$(2')$ а ббв ў«пов в ЄЁ¬ ®Ўа §®¬, зв®Ўл в®зЄҐ $A$ ᮮ⢥вбвў®ў «®
§­ зҐ­ЁҐ Ї а ¬Ґва  $t=a$,   в®зЄҐ $B$ -- §­ зҐ­ЁҐ $t=b$.\\ ‚ ¦­л¬
бў®©бвў®¬ ®ЇаҐ¤Ґ«Ґ­Ёп (2) пў«пҐвбп, Є Є Ј®ў®апв, ­Ґ§ ўЁбЁ¬®бвм
Їа ў®© з бвЁ (2) ®в ўлЎ®а  Ї а ¬ҐваЁ§ жЁЁ (1). ‘¬лб« нв®© да §л
б®бв®Ёв ў б«Ґ¤го饬: Їгбвм $t=u(v)$ -- ЇаҐ®Ўа §®ў ­ЁҐ ®в१Є 
$[c,d]$ ў ®в१®Є $[a,b]$, $u(\cdot)\in C^1[c,d]$: $$u'(v)\ne
0,\quad v\in [c,d],\ u(c)=a,\ u(d)=b.$$ ќв®, ў з бв­®бвЁ,
®§­ з Ґв, зв® $u(\cdot)$ -- бва®Ј® ¬®­®в®­­ п ­  $[c,d]$ дг­ЄжЁп.
ЋЎа §гҐ¬ б«®¦­го дг­ЄжЁо $p(v)=x(u(v))$ Ё ўлзЁб«Ё¬ ҐҐ Їа®Ё§ў®¤­го
ў в®зЄҐ $v\in (c,d)$. Џ® Їа ўЁ«г ¤ЁддҐаҐ­жЁа®ў ­Ёп б«®¦­®© дг­ЄжЁЁ
$$\frac{d\bigl(p(v)\bigr)}{dv}=\frac{d\bigl(x(u(v))\bigr)}{dv}=
\left.\frac{d\bigl(x(t)\bigr)}{dt}\right|_{t=u(v)}\cdot
\frac{d\bigl(u(v)\bigr)}{dv}.$$ Ђ­ «®ЈЁз­ п д®а¬г«  бЇа ўҐ¤«Ёў  Ё
¤«п дг­ЄжЁЁ $q(v)=y(u(v))$. ’ ЄЁ¬ ®Ўа §®¬,
$$\left.\sqrt{\left(\frac{d\bigl(x(t)\bigr)}{dt}\right)^2
+\left(\frac{d\bigl(y(t)\bigr)}{dt}\right)^2}\right|_{t=u(v)}=
\sqrt{\left(\frac{d\bigl(p(v)\bigr)}{dv}\right)^2
+\left(\frac{d\bigl(q(v)\bigr)}{dv}\right)^2}\cdot
\left|\frac{d\bigl(u(v)\bigr)}{dv}\right|^{-1}.$$ …б«Ё $c$ Ё $d$
а бЇ®«®¦Ґ­л ¤агЈ ®в­®бЁвҐ«м­® ¤агЈ  в Є¦Ґ Є Є Ё $a$ ®в­®бЁвҐ«м­®
$b$ (­ ЇаЁ¬Ґа, $c<d$ Ё ®¤­®ўаҐ¬Ґ­­® $a<b$), в®, ў бЁ«г
¬®­®в®­­®бвЁ $u(\cdot)$, $$\frac{d\bigl(u(v)\bigr)}{dv}>0, \quad
u\in[c,d].$$ …б«Ё ¦Ґ а бЇ®«®¦Ґ­ЁҐ нвЁе Ї а Їа®вЁў®Ї®«®¦­®
(­ ЇаЁ¬Ґа, $c>d$ Ё $a<b$), в® нв  Їа®Ё§ў®¤­ п ўбо¤г ¬Ґ­миҐ ­г«п.
Џ®н⮬г, ®Ў®§­ з п бЁ¬ў®«®¬ $\int_{[c,d]}$ Ё­вҐЈаЁа®ў ­ЁҐ Ї®
$[c,d]$ ®в ¬Ґ­м襣® §­ зҐ­Ёп  аЈг¬Ґ­в  Є Ў®«м襬㠨 ¤Ґ« п § ¬Ґ­г
ЇҐаҐ¬Ґ­­®© ў Їа ў®© з бвЁ (2), Ї®«гзЁ¬ (ЇаҐ¤Ї®« Ј Ґ¬, Є ЇаЁ¬Ґаг,
$a<b$): $$\int_a^b f(x(t),y(t)) \sqrt{{x'}^2(t)+{y'}^2(t)}\,dt=$$
$$=\int_c^d f(x(u(v)),y(u(v)))
\sqrt{\left(\frac{d\bigl(p(v)\bigr)}{dv}\right)^2
+\left(\frac{d\bigl(q(v)\bigr)}{dv}\right)^2}\cdot
\left|\frac{d\bigl(u(v)\bigr)}{dv}\right|^{-1}
\cdot\frac{d\bigl(u(v)\bigr)}{dv}\,dv=$$ $$=\int_{[c,d]}
f(x(u(v)),y(u(v)))
\sqrt{\left(\frac{d\bigl(g(v)\bigr)}{dv}\right)^2
+\left(\frac{d\bigl(h(v)\bigr)}{dv}\right)^2}\,dv.$$ ’®з­® в Є¦Ґ
а §ЎЁа Ґвбп б«гз © $a>b$. ’ ЄЁ¬ ®Ўа §®¬,
$$\int_{[a,b]}f(x(t),y(t))\sqrt{{x'}^2(t)+{y'}^2(t)}\,dt=
\int_{[c,d]} f(p(v),q(v))\sqrt{{g'}^2(v)+{h'}^2(v)}\,dv, \quad
(3)$$ Ё, Є Є Ўл ¬л ­Ё § ¤ ў «Ё ЄаЁўго $\gamma$, -- ў ўЁ¤Ґ $(1)$
Ё«Ё Є Є $$\gamma=\{(p(v),q(v))\ |\ v\in [c,d]\},$$ Їа ў п з бвм
(2) ­Ґ ¬Ґ­пҐв бў®ҐЈ® §­ зҐ­Ёп (3). ‚ н⮬ Ё б®бв®Ёв ­Ґ§ ўЁбЁ¬®бвм
Ё­вҐЈа «  ЇҐаў®Ј® த  ®в ўлЎ®а  Ї а ¬ҐваЁ§ жЁЁ Ё бЇ®б®Ў ҐЈ®
§ ЇЁбЁ $$\int_{\gamma}f(x,y)\,ds.$$ „«п ЄаЁў®«Ё­Ґ©­®Ј® Ё­вҐЈа « 
ўв®а®Ј® த , б®Ј« б­® ®ЇаҐ¤Ґ«Ґ­Ёо, ў ¦­®Ґ §­ зҐ­ЁҐ Ё¬ҐҐв Ї®а冷Є
Їа®е®¦¤Ґ­Ёп ўҐаиЁ­ $A$ Ё $B$: $$\int_{AB}f(x,y)\,dx=
\int_a^bf(x(t),y(t))x'(t)\,dt= \int_c^d f(x(u(v)),y(u(v)))
\frac{d\bigl(g(v)\bigr)}{dv}
\left(\frac{d\bigl(u(v)\bigr)}{dv}\right)^{-1}
\frac{d\bigl(u(v)\bigr)}{dv}\,dv=$$ $$=\int_c^d f(p(v),q(v))
g'(v)\,dv$$ Ё, ў ⮦Ґ ўаҐ¬п, $$\int_{BA}f(x,y)\,dx=
\int_b^af(x(t),y(t))x'(t)\,dt= \int_d^c f(p(v),q(v)) g'(v)\,dv=
-\int_{AB}f(x,y)\,dx.$$ ’®з­® в Є¦Ґ $$\int_{AB}g(x,y)\,dy=
-\int_{BA}g(x,y)\,dy$$ Ё, §­ зЁв,
$$\int_{{AB}}f(x,y)\,dx+g(x,y)\,dy=-
\int_{{BA}}f(x,y)\,dx+g(x,y)\,dy.$$ ’ Є¦Ґ Є Є Ё Ё­вҐЈа «л ЇҐаў®Ј®
த , Ё­вҐЈа «л ўв®а®Ј® த  ­Ґ ¬Ґ­пов бў®ҐЈ® §­ зҐ­Ёп ЇаЁ
ЇҐаҐЇ а ¬ҐваЁ§ жЁпе, б®еа ­пойЁе Ї®а冷Є Їа®е®¦¤Ґ­Ёп ўҐаиЁ­ $A$ Ё
$B$.

Џгбвм ⥯Ґам ЄаЁў п $\gamma\subset\mathbb{R}^2$ б®бв®Ёв Ё§
­ҐбЄ®«мЄЁе з б⥩ а бᬮв७­®Ј® ўЁ¤ :
$$\gamma=\cup_{i=1}^n\gamma_i,\quad \gamma_i=\{(x_i(t),y_i(t))\ |\
t\in [a_i,b_i]\},\eqno (4)$$ ЇаЁзҐ¬ $x_i(\cdot)$, $y_i(\cdot)\in
C[a_i,b_i]\cap C^1(a_i,b_i)$ Ё ЇаЁ ўбҐе $i$
$${x'}_i^2(t)+{y'}_i^2(t)>0,\quad t\in (a_i,b_i),$$ Їгбвм Є Є Ё
ЇаҐ¦¤Ґ а §­л¬ §­ зҐ­Ёп¬ Ї а ¬Ґва  $t_1\in(a_i,b_i)$,
$t_2\in(a_k,b_k)$ ᮮ⢥вбвўгов а §­лҐ в®зЄЁ
$(x_i(t_1),y_i(t_1))$, $(x_k(t_2),y_k(t_2))$ ЄаЁў®© $\gamma$.
‘®ў®ЄгЇ­®бвм в ЄЁе ЄаЁўле Ўг¤Ґ¬ ®Ў®§­ з вм §­ Є®¬ $\mathcal{A}_2$.
ЋЎ®§­ зЁ¬ $A_i=(x_i(a_i),y_i(a_i))$, $B_i=(x_i(b_i),y_i(b_i))$,
$i=1,\dots, n$. ќвЁ в®зЄЁ Ўг¤Ґ¬ ­ §лў вм 㧫 ¬Ё ЄаЁў®© $\gamma$.
…б«Ё $f(\cdot)$, $g(\cdot)\in C(\gamma)$ -- ¤ў 
¤Ґ©б⢨⥫쭮§­ з­ле ­ҐЇаҐалў­ле ®в®Ўа ¦Ґ­Ёп, в® Ї® ®ЇаҐ¤Ґ«Ґ­Ёо
$$\int_{\gamma}f\,ds=\sum_{i=1}^n\int_{\gamma_i}f\,ds,\eqno (5)$$
$$\int_{\gamma}f(x,y)\,dx+g(x,y)\,dy=
\sum_{i=1}^n\int_{A_i,B_i}f(x,y)\,dx+g(x,y)\,dy.\eqno (5')$$
ЉаЁўго $(4)$ Ўг¤Ґ¬ ­ §лў вм § ¬Є­гв®©, Ґб«Ё $B_1=A_2$,
$B_2=A_3$,\dots $B_{n-1}=A_n$ Ё $B_n=A_1$.

ђ бб¬ ваЁў ов в Є¦Ґ ЄаЁў®«Ё­Ґ©­лҐ Ё­вҐЈа «л ¤«п § ¤ ­­ле
Ї а ¬ҐваЁзҐбЄЁ Їа®бва ­б⢥­­ле ЄаЁўле
$$\gamma=\{(x(t),y(t),z(t))\ |\ t\in [a,b]\},\quad
A=(x(a),y(a),z(a)),\quad B=(x(b),y(b),z(b)).$$ “б«®ўЁҐ
­Ґўл஦¤Ґ­­®бвЁ ¤«п в ЄЁе ЄаЁўле § ЇЁблў ов в Є:
$${x'}^2(t)+{y'}^2(t)+{z'}^2(t)>0,\quad t\in (a,b).$$ Љ Є Ё ЇаҐ¦¤Ґ
бзЁв Ґ¬, зв® а §­л¬ §­ зҐ­Ёп¬ Ї а ¬Ґва  $t\in(a,b)$ ᮮ⢥вбвўгов
а §­лҐ в®зЄЁ $(x(t),y(t),z(t))$, б®бв ў«пойЁҐ ЄаЁўго $\gamma$.
ЋЇаҐ¤Ґ«Ґ­Ёп (2), $(2')$ ЇаЁ®ЎаҐв ов ўЁ¤
$$\int_{\gamma}f(x,y,z)\,ds=
\int_{[a,b]}f(x(t),y(t),z(t))\sqrt{{x'}^2(t)+{y'}^2(t)+{z'}^2(t)}\,dt,$$
$$\int_{AB}f(x,y,z)\,dx+g(x,y,z)\,dy+h(x,y,z)\,dz=
\int_a^b(f(t)x'(t)+g(t)y'(t)+h(t)z'(t))\,dt,$$ Ј¤Ґ
$f(t)=f(x(t),y(t),z(t))$ Ё в.¤. ЋЇаҐ¤Ґ«Ґ­Ёп (5), $(5')$
ЇҐаҐ­®бЁвбп ЎҐ§ Ё§¬Ґ­Ґ­Ё© (Є« бб ЄаЁўле, пў«пойЁебп  ­ «®Ј®¬
$\mathcal{A}_2$, Ўг¤Ґ¬ ®Ў®§­ з вм $\mathcal{A}_3$). ’ Є¦Ґ Є Є Ё
¤«п б«гз п ¤ўге ЇҐаҐ¬Ґ­­ле, Ё­вҐЈа « ЇҐаў®Ј® த  Ї®
Їа®бва ­б⢥­­л¬ ЄаЁўл¬ ­Ґ § ўЁбЁв ®в ўлЎ®а  Ї а ¬ҐваЁ§ жЁЁ,  
Ё­вҐЈа « ўв®а®Ј® த  § ўЁбЁв ®в ®аЁҐ­в жЁЁ Їа®бва ­б⢥­­®©
ЄаЁў®© (в.Ґ. ®в ўлЎ®а  ­ з «м­®© Ё Є®­Ґз­®© в®зЄЁ нв®© ЄаЁў®©):
$$\int_{AB}f(x,y,z)\,dx+g(x,y,z)\,dy+h(x,y,z)\,dz=
-\int_{BA}f(x,y,z)\,dx+g(x,y,z)\,dy+h(x,y,z)\,dz.$$ „«п б®Єа йҐ­Ёп
§ ЇЁбҐ© ўбҐ ¤ «м­Ґ©иЁҐ а бᬮв७Ёп Ўг¤гв ¤Ґ« вмбп ¤«п Ї«®бЄЁе
ЄаЁўле, ­® ®­Ё Ё¬Ґов ҐбвҐб⢥­­лҐ  ­ «®ЈЁ ¤«п Ё­вҐЈа «®ў Ї®
Їа®бва ­б⢥­­л¬ ЄаЁўл¬.

1. Џгбвм ЄаЁў п $\gamma$ § ¤ ­  га ў­Ґ­ЁҐ¬ $y=y(x)$, $x\in[a,b]$,
$y(\cdot)\in C^1[a,b]$. ќв® ®§­ з Ґв, зв® $\gamma=\{(t,y(t))\ |\
t\in [a,b]\}$. ‚ н⮬ б«гз Ґ (2), $(2')$ ЇаҐўа й овбп ў
$$\int_{\gamma}f(x,y)\,ds=
\int_a^bf(t,y(t))\sqrt{1+{y'}^2(t)}\,dt,$$
$$\int_{{AB}}f(x,y)\,dx+g(x,y)\,dy=
\int_a^b(f(t,y(t))+g(t,y(t))y'(t))\,dt.$$

2. ‘ў®©бвў  ЄаЁў®«Ё­Ґ©­ле Ё­вҐЈа «®ў:\\  )  ¤¤ЁвЁў­®бвм (Ї®
®Ў« бвЁ Ё­вҐЈаЁа®ў ­Ёп): Їгбвм ЄаЁўлҐ $\gamma_1$, $\gamma_2\in
\mathcal{A}_2$ Ё«Ё ­Ґ Ё¬Ґов ®ЎйЁе в®зҐЄ, Ё«Ё ®ЎйЁҐ в®зЄЁ нвЁе
ЄаЁўле Ї®Ї ¤ ов ­  Ёе 㧫л. ’®Ј¤  $\gamma_1\cup\,\gamma_2\in
\mathcal{A}_2$ Ё $$\int_{\gamma_1\cup\,\gamma_2}
f\,ds=\int_{\gamma_1} f\,ds+\int_{\gamma_2} f\,ds,\quad f\in
C(\gamma_1\cup\,\gamma_2),$$ $$\int_{\gamma_1\cup\,\gamma_2}
f\,dx+g\,dy=\int_{\gamma_1}f\,dx+g\,dy + \int_{\gamma_2}
f\,dx+g\,dy,\quad f,g\in C(\gamma_1\cup\,\gamma_2),$$ Ў)
«Ё­Ґ©­®бвм (Ї® Ё­вҐЈаЁагҐ¬л¬ дг­ЄжЁп¬):  Ґб«Ё $\gamma\in
\mathcal{A}_2$, $\ f_1,f_2,g_1,g_2\in C(\gamma)$,
$a,b\in\mathbb{R}$, в® $$\int_{\gamma}(af_1+bf_2)\,ds=a
\int_{\gamma}f_1\,ds+b\int_{\gamma}f_2\,ds,$$ $$\int_{\gamma}
(af_1+bf_2)\,dx+(ag_1+bg_2)\,dy= a\int_{\gamma}f_1\,dx+g_1\,dy
+b\int_{\gamma}f_2\,dx+g_2\,dy,$$ ў) Ї®«®¦ЁвҐ«м­®бвм (Ї®
Ё­вҐЈаЁагҐ¬л¬ дг­ЄжЁп¬): Їгбвм $\gamma\in \mathcal{A}_2$, $\
f_1,f_2\in C(\gamma)$. ’®Ј¤  $$\int_{\gamma}f_1\,ds\leqslant
\int_{\gamma}f_2\,ds,$$ Ґб«Ё $f_1(w)\leqslant f_2(w)$ ЇаЁ Є ¦¤®¬
$w\in\gamma$. „«п Ё­вҐЈа «®ў ўв®а®Ј® த   ­ «®ЈЁз­®Ґ бў®©бвў®,
ў®®ЎйҐ Ј®ў®ап, ­Ґ ўлЇ®«­пҐвбп.

„®Є § вҐ«мбвў® нвЁе бў®©бвў Їа®ў®¤Ёвбп ­  ®б­®ўҐ ®ЇаҐ¤Ґ«Ґ­Ё© (2),
$(2')$, (5), $(5')$ Ё Ё§ўҐбв­ле бў®©бвў ®ЇаҐ¤Ґ«Ґ­­®Ј® Ё­вҐЈа « .
Џа®ўҐаЁ¬, Є ЇаЁ¬Ґаг, Ї®«®¦ЁвҐ«м­®бвм Ё­вҐЈа « , ®бв «м­лҐ бў®©бвў 
гбв ­ ў«Ёў овбп Ї®  ­ «®ЈЁЁ. Џгбвм б­ з «  $\gamma$ -- ЄаЁў п ўЁ¤ 
(1). ’®Ј¤  Ї® (2) Ё бў®©бвў ¬ ®ЇаҐ¤Ґ«Ґ­­®Ј® Ё­вҐЈа « 
$$\int_{\gamma}f_1\,ds=\int_{[a,b]}f_1(x(t),y(t))
\sqrt{{x'}^2(t)+{y'}^2(t)}\,dt\leqslant\int_{[a,b]}f_2(x(t),y(t))
\sqrt{{x'}^2(t)+{y'}^2(t)}\,dt=\int_{\gamma}f_2\,ds$$ --
­Ґа ўҐ­бвў® бЇа ўҐ¤«Ёў®. …б«Ё ¦Ґ ЄаЁў п $\gamma$ б®бв®Ёв Ё§
­ҐбЄ®«мЄЁе гз бвЄ®ў а бᬮв७­®Ј® ўЁ¤ 
($\gamma=\cup_{i=1}^n\gamma_i$), в® Ї® (5) Ё в®«мЄ® зв®
гбв ­®ў«Ґ­­®¬г ­Ґа ўҐ­бвўг
$$\int_{\gamma}f_1\,ds=\sum_{i=1}^n\int_{\gamma_i}f_1\,ds\leqslant
\sum_{i=1}^n\int_{\gamma_i}f_2\,ds =\int_{\gamma}f_2\,ds.$$ Ј)
…б«Ё $f(\cdot)$ -- дг­ЄжЁп ⮦¤Ґб⢥­­® а ў­ п 1 (в.Ґ. а ў­ п 1
ЇаЁ ўбҐе §­ зҐ­Ёпе  аЈг¬Ґ­в ), в® $$l(\gamma)=\int_{\gamma}f\,ds=
\int_{\gamma}\,ds$$ ­ §лў ов ¤«Ё­®© ЄаЁў®© $\gamma$.\\ ¤)
(б«Ґ¤бвўЁҐ ў)) Џгбвм $m\leqslant f(w)\leqslant M$ ¤«п Є ¦¤®© в®зЄЁ
$w\in\gamma$. ’®Ј¤  $$m\cdot l(\gamma)\leqslant
\int_{\gamma}f\,ds\leqslant M\cdot l(\gamma),\quad
|\int_{\gamma}f\,ds|\leqslant\int_{\gamma}|f|\,ds.$$ ЏаЁ
а бᬮв७ЁЁ § ¤ з, бўп§ ­­ле б Ё­вҐЈаЁа®ў ­ЁҐ¬ дг­ЄжЁ© ­ҐбЄ®«мЄЁе
ЇҐаҐ¬Ґ­­ле, б«®¦Ё« бм ®ЇаҐ¤Ґ«Ґ­­ п вҐа¬Ё­®«®ЈЁп. „ ¤Ё¬ ­ҐЄ®в®алҐ
®ЇаҐ¤Ґ«Ґ­Ёп. Џгбвм $T\in\mathcal{T}_2$ (Ё«Ё  $T\in\mathcal{T}_3$),
$M=(x,y)\in T$ -- в®зЄ  ®Ў« бвЁ $T$ (Ё«Ё  $M=(x,y,z)$). ’®Ј¤ 
ўбпЄго дг­ЄжЁо $f(\cdot):T\to\mathbb{R}$ ­ §лў ов ҐйҐ бЄ «па­л¬
Ї®«Ґ¬, § ¤ ­­л¬ ­  ®Ў« бвЁ $T$. ‡­ зҐ­Ёп нв®Ј® Ї®«п ў в®зЄ е $M\in
T$ § ЇЁблў ов Є Є $f(M)$. …б«Ё $T\in\mathcal{T}_2$, $f(\cdot)$,
$g(\cdot):T\to\mathbb{R}$, Ї аг $a(\cdot)=(f(\cdot),g(\cdot))$
­ §лў ов ўҐЄв®а­л¬ Ї®«Ґ¬, § ¤ ­­л¬ ­  ®Ў« бвЁ $T$. …б«Ё
$T\in\mathcal{T}_3$, ўҐЄв®а­л¬ Ї®«Ґ¬ $a(\cdot)$, § ¤ ­­л¬ ­  $T$,
­ §лў ов ва®©Єг бЄ «па­ле Ї®«Ґ© ­  $T$:
$a(\cdot)=(f(\cdot),g(\cdot),h(\cdot))$. ‡­ зҐ­Ёп нвЁе Ї®«Ґ© ў
в®зЄ е $M\in T$ § ЇЁблў ов Є Є $a(M)$. Ѓг¤Ґ¬ § ЇЁблў вм
$f(\cdot)\in C^1(T)$, Ґб«Ё бЄ «па­®Ґ Ї®«Ґ $f(\cdot)$ Ё ҐЈ® з бв­лҐ
Їа®Ё§ў®¤­лҐ ®ЇаҐ¤Ґ«Ґ­л Ё ­ҐЇаҐалў­л ў Є ¦¤®© в®зЄҐ $M\in T$. Ѓг¤Ґ¬
§ ЇЁблў вм $a(\cdot)\in C^1(T)$, Ґб«Ё Є®®а¤Ё­ вл нв®Ј® ўҐЄв®а­®Ј®
Ї®«п $a=(a_x,a_y)$ (Ё«Ё $a=(a_x,a_y,a_z)$ ў ваҐе¬Ґа­®¬ б«гз Ґ)
ЇаЁ­ ¤«Ґ¦ в Є« ббг $C^1(T)$. Џгбвм $f(\cdot)\in C^1(T)$ --
бЄ «па­®Ґ Ї®«Ґ. ѓа ¤ЁҐ­в®¬ $f(\cdot)$ ­ §лў ов ўҐЄв®а­®Ґ Ї®«Ґ
${\rm grad}\,f$, Є®¬Ї®­Ґ­вл Є®в®а®Ј® -- з бв­лҐ Їа®Ё§ў®¤­лҐ
$f(\cdot)$: $${\rm grad}\,f=\left(\frac{\partial f}{\partial
x},\frac{\partial f}{\partial y}\right)\quad ({\rm Ё«Ё}\ {\rm
grad}\,f=\left(\frac{\partial f}{\partial x},\frac{\partial
f}{\partial y},\frac{\partial f}{\partial z}\right)).$$ Џгбвм
⥯Ґам $\gamma$ -- Ї«®бЄ п Ё«Ё Їа®бва ­б⢥­­ п § ¬Є­гв п ЄаЁў п
ўЁ¤  (4), $a(\cdot)\in C(\gamma)$ -- ­ҐЇаҐалў­®Ґ ўҐЄв®а­®Ґ Ї®«Ґ ­ 
$\gamma$. €­вҐЈа « (¤«п Ї«®бЄ®© ЄаЁў®© $a_z(M)=0$)
$$\int_{\gamma}a_x\,dx+a_y\,dy+a_z\,dz$$ ­ §лў Ґвбп жЁаЄг«пжЁҐ©
ўҐЄв®а­®Ј® Ї®«п $a(\cdot)$ Ї® ЄаЁў®© $\gamma$. Ћ­ в Є¦Ґ Є®а®вЄ®
®Ў®§­ з Ґвбп $$\int_{\gamma}a\cdot dr,$$ Ј¤Ґ $dr=(dx,dy,dz)$,
$a\cdot dr$ -- бЄ «па­®Ґ Їа®Ё§ўҐ¤Ґ­ЁҐ ўҐЄв®а®ў (ў Ї«®бЄ®¬ б«гз Ґ
-- ўбҐ ўҐЄв®а  ¤ўг¬Ґа­лҐ). Ћв¬ҐвЁ¬ ҐйҐ бўп§м ¬Ґ¦¤г а бᬮв७­л¬Ё
ЄаЁў®«Ё­Ґ©­л¬Ё Ё­вҐЈа « ¬Ё. Џгбвм $$\gamma=\{(x(t),y(t))\ |\ t\in
[a,b]\}$$ -- Ї а ¬ҐваЁзҐбЄ®Ґ ЇаҐ¤бв ў«Ґ­ЁҐ ЄаЁў®© $\gamma$ ­ 
Ї«®бЄ®бвЁ ($x,y\in C^1(a,b)$). ‚ҐЄв®а $(x'(t),y'(t))$, $t\in
(a,b)$, ҐбвҐб⢥­­® ­ §ў вм ­ Їа ў«Ґ­ЁҐ¬ Є б вҐ«м­®© Їаאַ©,
Їа®ўҐ¤Ґ­­®© ў в®зЄҐ $(x(t),y(t))$, Є Ї а ¬ҐваЁзҐбЄЁ § ¤ ­­®©
ЄаЁў®© $\gamma$ (Ї®пб­Ґ­ЁҐ: ўҐЄв®а $(x'(t),y'(t))$ ®в«®¦Ґ­ ®в
­ з «  Є®®а¤Ё­ в. ‘ ҐЈ® Ї®¬®ймо Є б вҐ«м­ п Є $\gamma$ ў в®зЄҐ
$t\in (a,b)$ § ¤ Ґвбп Ї а ¬ҐваЁзҐбЄЁ: $$X(s)=x(t)+s\cdot
x'(t),\quad Y(s)=y(t)+s\cdot y'(t), \quad s\in\mathbb{R}.)$$ ЏаЁ
а §«Ёз­ле ЇҐаҐЇ а ¬ҐваЁ§ жЁпе ­ Їа ў«пойЁҐ ўҐЄв®а , ®в­®бпйЁҐбп Є
®¤­®© Ё в®© ¦Ґ в®зЄҐ ЄаЁў®© $\gamma$, ¬®Јгв Ё§¬Ґ­Ёвмбп Ї® ¤«Ё­Ґ
Ё«Ё Ё§¬Ґ­Ёвм ­ Їа ў«Ґ­ЁҐ ­  Їа®вЁў®Ї®«®¦­®Ґ. „агЈЁ¬Ё б«®ў ¬Ё в ЄЁҐ
­ Їа ў«пойЁҐ ўҐЄв®а  ўбҐЈ¤  «Ґ¦ в ­  ®¤­®© Їаאַ© (Ї а ««Ґ«м­®©
Є б вҐ«м­®© Ё Їа®е®¤п饩 зҐаҐ§ ­ з «® Є®®а¤Ё­ в), ЇаЁзҐ¬, ў бЁ«г
­ иЁе Ёб室­ле ЇаҐ¤Ї®«®¦Ґ­Ё© ® Ї а ¬ҐваЁ§ жЁЁ ЄаЁў®© $\gamma$, ­Ё
®¤Ё­ Ё§ Є б вҐ«м­ле ўҐЄв®а®ў ­Ґ а ўҐ­ ­г«Ґў®¬г ўҐЄв®аг. Љ®®а¤Ё­ вл
­®а¬Ёа®ў ­­®Ј® ўҐЄв®а 
$$\left(\frac{x'(t)}{\sqrt{{x'}^2(t)+{y'}^2(t)}},
\frac{y'(t)}{\sqrt{{x'}^2(t)+{y'}^2(t)}}\right)=
(\cos\alpha,\sin\alpha)=(\cos\alpha,\cos\beta),\quad
\alpha=\alpha(t),$$ $\beta=\pi/2-\alpha$, ­ §лў ов ­ Їа ў«пойЁ¬Ё
Є®бЁ­гб ¬Ё Є б вҐ«м­®©. ‚ᥠнвЁ ®ЇаҐ¤Ґ«Ґ­Ёп ҐбвҐб⢥­­л¬ ®Ўа §®¬
ЇҐаҐ­®бпвбп ­  ®ЎйЁҐ ЄаЁўлҐ $\gamma\in\mathcal{A}_2$,   в Є¦Ґ ­ 
Їа®бва ­б⢥­­лҐ ЄаЁўлҐ $\gamma\in\mathcal{A}_3$. ’ Є Є®®а¤Ё­ вл
ўҐЄв®а 
$$\left(\frac{x'(t)}{\sqrt{{x'}^2(t)+{y'}^2(t)+{z'}^2(t)}},
\frac{y'(t)}{\sqrt{{x'}^2(t)+{y'}^2(t)+{z'}^2(t)}},
\frac{z'(t)}{\sqrt{{x'}^2(t)+{y'}^2(t)+{z'}^2(t)}}\right)=
(\cos\alpha,\cos\beta,\cos\delta)$$ ­ §лў ов ­ Їа ў«пойЁ¬Ё
Є®бЁ­гб ¬Ё Є б вҐ«м­®© Є Їа®бва ­б⢥­­®© ЄаЁў®©
$$\gamma=\{(x(t),y(t),z(t))\ |\ t\in [a,b]\},$$ ўлзЁб«Ґ­­л¬Ё ў
в®зЄҐ $t\in (a,b)$. ‚ а бᬮв७­ле вҐа¬Ё­ е ®ЇаҐ¤Ґ«Ґ­ЁҐ
ЄаЁў®«Ё­Ґ©­®Ј® Ё­вҐЈа «  ўв®а®Ј® த  Ї® Їа®бва ­б⢥­­®© ЄаЁў®©
$AB$ ¬®¦Ґв Ўлвм ᤥ« ­® б«Ґ¤гойЁ¬ ®Ўа §®¬. Џгбвм $a=(f,g,h)$ --
­ҐЇаҐалў­®Ґ ўҐЄв®а­®Ґ Ї®«Ґ ў¤®«м ЄаЁў®© $AB$, $dr=(dx,dy,dz)$,
Їгбвм $\tau=(\cos\alpha,\cos\beta,\cos\delta)$ -- ­ Їа ў«пойЁҐ
Є®бЁ­гбл Є б вҐ«м­®© Є нв®© ЄаЁў®©. ’®Ј¤  $$\int_{AB}a\cdot
dr=\int_{AB}a\cdot\tau \,ds,$$ Ј¤Ґ
$a\cdot\tau=f\cos\alpha+g\cos\beta+h\cos\delta$ -- бЄ «па­®Ґ
Їа®Ё§ўҐ¤Ґ­ЁҐ ўҐЄв®а®ў.\\ „ «м­Ґ©иЁҐ Ї®бв஥­Ёп ®в­®бпвбп Є  ­ «®Јг
д®а¬г«л Ќмов®­ -‹Ґ©Ў­Ёж  ¤«п ¤ў®©­®Ј® Ё­вҐЈа « . Џгбвм
$T\in\mathcal{T}_2$ Ё¬ҐҐв ўЁ¤ $T=Y(\varphi,\psi)$ б Ј« ¤ЄЁ¬Ё
дг­ЄжЁп¬Ё $\varphi,\psi\in C^1(a,b)$, 㤮ў«Ґвў®апойЁ¬Ё
­Ґа ўҐ­бвў ¬ $$\varphi(t)<\psi(t),\quad t\in (a,b),\quad a<b.$$
ѓа ­ЁжҐ© $\partial T$ ¬­®¦Ґбвў  $T$, Ўг¤Ґ¬ ­ §лў вм § ¬Є­гвго
ЄаЁўго, ®Є ©¬«пойго $T$ Ё б®бв®пйго Ё§ ­Ґ Ў®«ҐҐ, 祬 зҐвлаҐе
гз бвЄ®ў. ќвЁ гз бвЄЁ § ¤ овбп Ї а ¬ҐваЁзҐбЄЁ Ї® Їа ўЁ« ¬:

1. $\gamma_1$ ®ЇаҐ¤Ґ«пҐвбп ­ Ў®а®¬ $x_1(t)=t$,
$y_1(t)=\varphi(t)$,\quad $t\in[a,b]$,

2. Ґб«Ё $\varphi(b)=\psi(b)$, в® бзЁв Ґ¬, зв® $\gamma_2$
®вбгвбвўгҐв, Ґб«Ё ¦Ґ $\varphi(b)<\psi(b)$, в® $\gamma_2$
®ЇаҐ¤Ґ«пҐвбп ­ Ў®а®¬ $x_2(t)=b$, $y_2(t)=t$,\quad
$t\in[\varphi(b),\psi(b)]$,

3. зв®Ўл ®ЎҐбЇҐзЁвм Ї®бв㯠⥫쭮Ґ ¤ўЁ¦Ґ­ЁҐ Ї® д®а¬Ёа㥬®¬г
Є®­вгаг, гз бв®Є $\gamma_3$ Ї а ¬ҐваЁ§гов б«Ґ¤гойЁ¬ ­ Ў®а®¬:
$x_3(t)=t$, $y_3(t)=\psi(t)$,\quad $t\in[b,a]$,

4. Ґб«Ё $\varphi(a)=\psi(a)$, в® бзЁв Ґ¬, зв® $\gamma_4$
®вбгвбвўгҐв, Ґб«Ё ¦Ґ $\varphi(a)<\psi(a)$, в® $\gamma_4$
®ЇаҐ¤Ґ«пҐвбп ­ Ў®а®¬ $x_4(t)=a$, $y_4(t)=t$,\quad
$t\in[\psi(a),\varphi(a)]$.

’ ЄЁ¬ ®Ўа §®¬, бд®а¬Ёа®ў «Ёбм г§«л Є®­вга  $\partial T$ Ё Ї®а冷Є
Ёе Їа®е®¦¤Ґ­Ёп: $B_1=A_2=(b,\varphi(b))$, $B_2=A_3=(b,\psi(b))$,
$B_3=A_4=(a,\psi(a))$, $B_4=A_1=(a,\varphi(a))$. …б«Ё ®¤Ё­ Ё«Ё ®Ў 
Ё§ гз бвЄ®ў $\gamma_2$, $\gamma_4$ ®вбгвбвўгҐв, в® ­ҐЄ®в®алҐ Ё§
нвЁе 㧫®ў б«Ёў овбп ў ®¤Ё­.

‚лЎа ­­л© Ї®а冷Є ®Ўе®¤  Є®­вга  $\partial T^+=
\{A_1,A_2,A_3,A_4,A_1\}$ ­ §лў ов ҐйҐ ®Ўе®¤®¬ ў Ї®«®¦ЁвҐ«м­®¬
­ Їа ў«Ґ­ЁЁ Ё«Ё ®Ўе®¤®¬ Їа®вЁў з б®ў®© бв५ЄЁ. Ћ­ е а ЄвҐаЁ§гҐвбп
⥬, зв® ЇаЁ гЄ § ­­®¬ ¤ўЁ¦Ґ­ЁЁ ў¤®«м $\partial T$, ®Ў« бвм $T$
®бв Ґвбп б«Ґў  ®в "®Ўе®¤зЁЄ ". ЉаЁўго $\partial T^+$, ЇаЁ н⮬,
­ §лў ов Ї®«®¦ЁвҐ«м­® ®аЁҐ­вЁа®ў ­­®©, нвг ¦Ґ ЄаЁўго, ­®
Їа®е®¤Ё¬го ў ®Ўа в­®¬ ­ Їа ў«Ґ­ЁЁ ®Ў®§­ з ов $\partial T^-$ Ё
­ §лў ов ®ваЁж вҐ«м­® ®аЁҐ­вЁа®ў ­­®©.

Ћв¬ҐвЁ¬ §¤Ґбм ҐйҐ ­Ґўл஦¤Ґ­­®бвм ўлЎа ­­®© Ї а ¬ҐваЁ§ жЁЁ Є®­вга 
$\partial T^+$:\\ 1. ў б«гз Ґ $\gamma_1$ ўлЇ®«­пҐвбп
$(t')^2+(\varphi'(t))^2=1+(\varphi'(t))^2>0$,\\ 2. Ґб«Ё $\gamma_2$
ЇаЁбгвбвўгҐв, в® $(b')^2+(t')^2=0+1>0$ Ё в.¤.

Џгбвм ⥯Ґам $g\in C(T)$ ®Ў« ¤ Ґв ­ҐЇаҐалў­®© з бв­®© Їа®Ё§ў®¤­®©
Ї® ЇҐаҐ¬Ґ­­®© $y$ ў ®Ў« бвЁ $T$: $\frac{\partial g}{\partial y}\in
C(T)$. ЏаҐ®Ўа §гҐ¬ ўла ¦Ґ­ЁҐ $$\int\!\!\int_T\frac{\partial g}
{\partial y}\,dx\,dy= \int_a^b\left( \int_{\varphi(x)}
^{\psi(x)}\frac{\partial g} {\partial y}\,dy\right)dx=
\int_a^b\bigl( g(x,\psi(x))-g(x,\varphi(x))\bigr)dx= \int_a^b
g(t,\psi(t))t'\,dt-$$
$$-\int_a^bg(t,\varphi(t))t'\,dt=\int_{B_3A_3}g(x,y)\,dx-
\int_{A_1B_1}g(x,y)\,dx=-\int_{A_4B_4}g(x,y)\,dx-$$
$$-\int_{A_3B_3}g(x,y)\,dx -\int_{A_2B_2}g(x,y)\,dx-
\int_{A_1B_1}g(x,y)\,dx=-\int_{\partial T^+}g(x,y)\,dx.$$ Џ® 室г
ЇаҐ®Ўа §®ў ­Ё© ¬л ў®бЇ®«м§®ў «Ёбм а ўҐ­бвў ¬Ё
$$\int_{A_4B_4}g(x,y)\,dx=\int_{\psi(a)}^{\varphi(a)}
g(a,t)a'\,dt= 0,\quad \int_{A_2B_2}g(x,y)\,dx=0,$$ -- ў б«гз Ґ,
Є®Ј¤  ®¤­®Ј® Ё«Ё ®Ў®Ёе Ё§ нвЁе гз бвЄ®ў $\gamma^+$ ­Ґв, в®
ᮮ⢥вбвўгойЁҐ Ё¬ б« Ј Ґ¬лҐ, Ї® ®ЇаҐ¤Ґ«Ґ­Ёо, бзЁв Ґ¬ а ў­л¬Ё
­г«о. ’ ЄЁ¬ ®Ўа §®¬, ®Є®­з вҐ«м­ п д®а¬г«  Ё¬ҐҐв ўЁ¤
$$\int\!\!\int_T\frac{\partial g} {\partial
y}\,dx\,dy=-\int_{\partial T^+}g(x,y)\,dx.\eqno (6)$$ Џгбвм ⥯Ґам
$T\in\mathcal{T}_2$ Ё¬ҐҐв Ў®«ҐҐ б«®¦­го бвагЄвгаг:
$$T=\bigcup_{i=1}^nT_i,\quad \widehat{T}_i\cap
\widehat{T}_j=\emptyset, i\ne j,$$ Ё Є ¦¤®Ґ Ё§ $T_i$ Ё¬ҐҐв ўЁ¤
$Y(\varphi_i,\psi_i)$ б Ј« ¤ЄЁ¬Ё дг­ЄжЁп¬Ё $\varphi_i,\psi_i\in
C^1(a_i,b_i)$ бў®Ё¬Ё ¤«п Є ¦¤®Ј® $i$ Ё 㤮ў«Ґвў®апойЁ¬Ё
­Ґа ўҐ­бвў ¬ $$\varphi_i(t)<\psi_i(t),\quad t\in (a_i,b_i),\quad
a_i<b_i,\quad i=1,\dots, n.$$ ‘д®а¬г«Ёа㥬 ҐйҐ ®¤­® ®Ја ­ЁзҐ­ЁҐ ­ 
бвагЄвгаг $T$. „«п нв®Ј® ®Ўа §гҐ¬ ¬­®¦Ґбвў® $\partial T$ Ё Ўг¤Ґ¬
­ §лў вм ҐЈ® Ја ­ЁжҐ© $T$ Ї® б«Ґ¤го饬㠯ࠢЁ«г: Їгбвм $\partial
T_i^+$ -- Ја ­Ёжл ¬­®¦Ґбвў $T_i$, $i\in\{1,\dots, n \}$,
®аЁҐ­вЁа®ў ­­лҐ Їа®вЁў з б®ў®© бв५ЄЁ. ’®Ј¤  $(x,y)\in\partial
T$, Ґб«Ё $(x,y)\in\partial T_i$ ў в®з­®бвЁ ¤«п ®¤­®Ј® Ё§
$i\in\{1,\dots, n \}$. (—Ёв о饬г ४®¬Ґ­¤гҐвбп ᤥ« вм ­ҐбЄ®«мЄ®
аЁбг­Є®ў ў®§¬®¦­®Ј® бв஥­Ёп $T$ Ё $\partial T$ ¤«п $n=2$ Ё
$n=3$.) ЋЈа ­ЁзҐ­ЁҐ б®бв®Ёв ў ⮬, зв® ¬­®¦Ґбвў® $\partial T$,
Є®в®а®Ґ ў®®ЎйҐ Ј®ў®ап ¬®¦Ґв Ўлвм гбв஥­® ¤®бв в®з­® б«®¦­®,
Ї®¤а §¤Ґ«пҐвбп ­  Є®­Ґз­®Ґ зЁб«® ­ҐЇҐаҐбҐЄ ойЁебп ®в१Є®ў,
Ї®«г®в१Є®ў, Ё­вҐаў «®ў Ё ЄаЁўле ўЁ¤  $\{(t,\varphi(t))\ |\
t\in(a,b)\}$. ‚ᥠЇҐаҐзЁб«Ґ­­лҐ н«Ґ¬Ґ­вл, Є®­Ґз­®, ЇаЁ н⮬
пў«повбп з бвп¬Ё в®© Ё«Ё Ё­®© $\partial T_i$, $i\in\{1,\dots, n
\}$, Ёе ®аЁҐ­в жЁп Ё Ї а ¬ҐваЁ§ жЁп ®¤­®§­ з­® § ¤ овбп ўлЎ®а®¬
®аЁҐ­в жЁЁ Ё Ї а ¬ҐваЁ§ жЁЁ ᮮ⢥вбвўго饣® $\partial T_i$.
„®Ў ўЁ¬ Є $\partial T$ Ја ­Ёз­лҐ в®зЄЁ ўбҐе нвЁе Ї®«г®в१Є®ў,
Ё­вҐаў «®ў, в®зЄЁ ўЁ¤  $(a,\varphi(a))$, $(b,\varphi(b))$ Ё,
­Ґб¬®вап ­  в®, зв® ¬л ­Ґ¬­®Ј® Ё§¬Ґ­Ё«Ё ¬­®¦Ґбвў® $\partial T$,
б®еа ­Ё¬ §  Ё§¬Ґ­Ґ­­л¬ ­ §ў ­ЁҐ Ё ®Ў®§­ зҐ­ЁҐ. Њ®¦­® Ї®Є § вм, зв®
$\partial T\in \mathcal{A}_2$ Ё зв® Ї®«г祭­ п ЄаЁў п ­Ґ § ўЁбЁв
®в Ёб室­®Ј® бЇ®б®Ў  а §¤Ґ«Ґ­Ёп $T$ ­  з бвЁ. ’ Є Є Є $$\partial
T\subset\bigcup_{i=1}^n\partial T_i,$$ в® Ї а ¬ҐваЁ§ жЁп Ё Ї®а冷Є
Їа®е®¦¤Ґ­Ёп з б⥩ $\partial T$ Ї®«­®бвмо ®ЇаҐ¤Ґ«пҐвбп
Ї а ¬ҐваЁ§ жЁп¬Ё Ё Ї®ап¤Є ¬Ё Їа®е®¦¤Ґ­Ёп Ја ­Ёж $\partial T_i$.
Џ®«г祭­го ®аЁҐ­вЁа®ў ­­го ЄаЁўго Ўг¤Ґ¬ ®Ў®§­ з вм $\partial T^+$.
Ћ­  в Є¦Ґ ®Ў« ¤ Ґв ⥬ бў®©бвў®¬, зв® ЇаЁ ¤ўЁ¦Ґ­ЁЁ ў¤®«м $\partial
T$ ў ўлЎа ­­®¬ ­ Їа ў«Ґ­ЁЁ ®Ў« бвм $T$ ®бв Ґвбп б«Ґў  ®в
"®Ўе®¤зЁЄ ". Љ Є Ё ЇаҐ¦¤Ґ в Є®© ®Ўе®¤ Ја ­Ёжл ®Ў« бвЁ Ўг¤Ґ¬
­ §лў вм ®Ўе®¤®¬ ў Ї®«®¦ЁвҐ«м­®¬ ­ Їа ў«Ґ­ЁЁ Ё«Ё ®Ўе®¤®¬ Їа®вЁў
з б®ў®© бв५ЄЁ. Џа®вЁў®Ї®«®¦­л© ®Ўе®¤ ­ §лў ов ®Ўе®¤®¬ ў
®ваЁж вҐ«м­®¬ ­ Їа ў«Ґ­ЁЁ. ЋЄ §лў Ґвбп, ¤«п а бᬮв७­®Ј®
¬­®¦Ґбвў  $T$ Ё дг­ЄжЁ© $g$, $\frac{\partial g}{\partial y}\in
C(T)$ д®а¬г«  (6) ®бв Ґвбп бЇа ўҐ¤«Ёў®©. (‡¤Ґбм ў­®ўм
४®¬Ґ­¤гҐвбп ᤥ« вм ­ҐбЄ®«мЄ® аЁбг­Є®ў ў®§¬®¦­®Ј® бв஥­Ёп $T$ Ё
$\partial T$ ¤«п $n=2$ Ё $n=3$.) Џ® бгвЁ ¤Ґ«  Ё¬Ґ­­® нвг д®а¬г«г ў
ᤥ« ­­ле ЇаҐ¤Ї®«®¦Ґ­Ёпе ®в­®бЁвҐ«м­® $T$ Ё $g$ ¬®¦­® ­ §лў вм
д®а¬г«®© ѓаЁ­ . Ћ¤­ Є® ҐҐ ЇаЁ­пв® § ЇЁблў вм ў ­ҐбЄ®«мЄ® Ў®«ҐҐ
бЁ¬¬ҐваЁз­®¬ ўЁ¤Ґ. Ђ Ё¬Ґ­­®, Їгбвм ¬­®¦Ґбвў® $T$, Ї®¬Ё¬®
а бᬮв७­®Ј®, ¤®ЇгбЄ Ґв Є®­Ґз­®Ґ а §«®¦Ґ­ЁҐ ўЁ¤  $$T=\bigcup_j
S_j,\quad \widehat{S}_j\cap \widehat{S}_i=\emptyset, j\ne i,$$ Ё
Є ¦¤®Ґ Ё§ $S_j$ Ё¬ҐҐв ўЁ¤ $X(\varphi_j,\psi_j)$ б® ўбҐ¬Ё
Ї®б«Ґ¤гойЁ¬Ё ®Ја ­ЁзҐ­Ёп¬Ё Ё ®ЇаҐ¤Ґ«Ґ­Ёп¬Ё, ¤Ґ« ойЁ¬Ёбп Є Є Ё ўлиҐ
¤«п ¬­®¦Ґбвў $Y(\varphi_j,\psi_j)$. ’®Ј¤  ҐйҐ ®¤­® Ї®бв஥­ЁҐ
$\partial T$ б®ўЇ ¤Ґв б 㦥 Ё¬ҐойЁ¬бп Ё б®ўЇ ¤гв Ї®«®¦ЁвҐ«м­лҐ
®аЁҐ­в жЁЁ нвЁе Ја ­Ёж. (‡ ¬Ґз ­ЁҐ: Ї а ¬ҐваЁ§ жЁЁ Є®­вга 
$\partial T$ ЇаЁ н⮬ ®Є ¦гвбп а §«Ёз­л¬Ё, ­®, Є Є 㦥 Ўл«®
®в¬ҐзҐ­®, Ё­вҐЈа « ўв®а®Ј® த , гз бвўгойЁ© ў (6), ­Ґзгўб⢨⥫Ґ­
Є ЇҐаҐЇ а ¬ҐваЁ§ жЁп¬, б®еа ­пойЁ¬ ®аЁҐ­в жЁо ЄаЁўле.)

”®а¬г«®© ѓаЁ­  ¤«п дг­ЄжЁ© $f,g\in C(T)$, г Є®в®але бгйҐбвўгов
$\frac{\partial f}{\partial x},\,\frac{\partial g}{\partial y}\in
C(T)$, ЇаЁ­пв® ­ §лў вм а ўҐ­бвў®
$$\int\!\!\int_T\left(\frac{\partial f}{\partial x}
-\frac{\partial g} {\partial y}\right)\,dx\,dy=\int_{\partial
T^+}g(x,y)\,dx+f(x,y)\,dy.\eqno (7)$$ ђ §«ЁзЁҐ ў §­ Є е з б⥩
нв®© д®а¬г«л бўп§ ­® б а §«ЁзЁҐ¬ ў Їа®е®¦¤Ґ­ЁЁ Ја ­Ёжл $\partial
T^+$: ¤«п в®© з бвЁ д®а¬г«л, ў Є®в®а®© гз бвўгҐв дг­ЄжЁп $g$,
¤ўЁ¦Ґ­Ёо ў Ї®«®¦ЁвҐ«м­®¬ ­ Їа ў«Ґ­ЁЁ ᮮ⢥вбвўгҐв ¤ўЁ¦Ґ­ЁҐ Ї®
"­Ё¦­Ґ©" (®в­®бЁвҐ«м­® ­ Їа ў«Ґ­Ёп $Oy$) з бвЁ $\partial T^+$,
ЇаЁў®¤п饥 Є 㢥«ЁзҐ­Ёо Є®®а¤Ё­ вл $x$. Ђ ¤«п в®© з бвЁ, ў Є®в®а®©
гз бвўгҐв дг­ЄжЁп $f$, ¤ўЁ¦Ґ­Ёо ў Ї®«®¦ЁвҐ«м­®¬ ­ Їа ў«Ґ­ЁЁ
ᮮ⢥вбвўгҐв ¤ўЁ¦Ґ­ЁҐ Ї® "­Ё¦­Ґ©" (®в­®бЁвҐ«м­® ­ Їа ў«Ґ­Ёп $Ox$)
з бвЁ $\partial T^+$, ЇаЁў®¤п饥 Є 㬥­м襭Ёо Є®®а¤Ё­ вл $г$. „«п
§ Ї®¬Ё­ ­Ёп д®а¬г«л (7) ¬®¦­® ЁбЇ®«м§®ў вм ¬­Ґ¬®­ЁзҐбЄ®Ґ Їа ўЁ«®
$$\int\!\!\int_T\frac{\partial f}{\partial x}\,dx\wedge dy
+\frac{\partial g} {\partial y}\,dy\wedge dx=\int_{\partial
T^+}g(x,y)\,dx+f(x,y)\,dy,$$ Ј¤Ґ $dx\wedge dy$, $dy\wedge dx$ --
н«Ґ¬Ґ­вл ®аЁҐ­вЁа®ў ­­®Ј® ®ЎкҐ¬  (Ї«®й ¤Ё): $dx\wedge dy=-dy\wedge
dx= dx\,dy$. ‘д®а¬г«Ёа㥬 ҐйҐ а § ⥮६㠮 ᢥ¤Ґ­ЁЁ ¤ў®©­®Ј®
Ё­вҐЈа «  Є ЄаЁў®«Ё­Ґ©­®¬г. ЏаЁ н⮬ Ўг¤Ґ¬ бзЁв вм Ё­вгЁвЁў­®
Ї®­пв­л¬ вҐа¬Ё­ $\partial T^+$ - Ја ­Ёж  ¬­®¦Ґбвў  ’, Їа®е®¤Ё¬ п ў
Ї®«®¦ЁвҐ«м­®¬ ­ Їа ў«Ґ­ЁЁ (в.Ґ. в Є, зв® ЇаЁ ¤ўЁ¦Ґ­ЁЁ ў гЄ § ­­®¬
­ Їа ў«Ґ­ЁЁ ¬­®¦Ґбвў® $T$ ®бв Ґвбп б«Ґў ). Ѓг¤Ґ¬ бзЁв вм, зв® $T$
¬®¦Ґв Ўлвм а §¤Ґ«Ґ­® ­  Є®­Ґз­®Ґ зЁб«® ¬­®¦Ґбвў ўЁ¤ 
$Y(\varphi,\psi)$, ­Ґ Ё¬ҐойЁе ®ЎйЁе ў­гв७­Ёе в®зҐЄ Ё, Єа®¬Ґ
в®Ј®, ¬®¦Ґв Ўлвм а §¤Ґ«Ґ­® ­  Є®­Ґз­®Ґ зЁб«® ¬­®¦Ґбвў ўЁ¤ 
$X(\varphi,\psi)$, в Є¦Ґ ­Ґ Ё¬ҐойЁе ®ЎйЁе ў­гв७­Ёе в®зҐЄ (ўбо¤г
ЇаЁ в ЄЁе ¤Ґ«Ґ­Ёпе $\varphi,\psi$ -- ­ҐЇаҐалў­® ¤ЁддҐаҐ­жЁа㥬лҐ
дг­ЄжЁЁ).\\ ’Ґ®аҐ¬  (д®а¬г«  ѓаЁ­ ): /$T\in\mathcal{T}_2$, \
$f,g\in C^1(T)$, \ $\partial T^+\in\mathcal{A}_2$/ $\Rightarrow$
$$\int\!\!\int_T\left(\frac{\partial f}{\partial x}
-\frac{\partial g} {\partial y}\right)\,dx\,dy=\int_{\partial
T^+}g(x,y)\,dx+f(x,y)\,dy.$$
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