Теорема на Стокс

$\displaystyle\oint_\redE{C}\blueE{\textbf{F}} \cdot d \textbf{r}$\oint, start subscript, start color #bc2612, C, end color #bc2612, end subscript, start color #0c7f99, start bold text, F, end bold text, end color #0c7f99, dot, d, start bold text, r, end bold text

$\displaystyle\oint_\redE{C}\blueE{\textbf{F}} \cdot d \textbf{r}$\oint, start subscript, start color #bc2612, C, end color #bc2612, end subscript, start color #0c7f99, start bold text, F, end bold text, end color #0c7f99, dot, d, start bold text, r, end bold text

$\displaystyle \underbrace{ \oint_{\greenE{C_1}} \blueE{\textbf{F}} \cdot d\textbf{r} + \oint_{\goldE{C_2}} \blueE{\textbf{F}} \cdot d\textbf{r} }_{\text{унищожават се по протежение на сечението на $\redE{S}$}} = \oint_{\redE{C}} \blueE{\textbf{F}} \cdot d\textbf{r}$start underbrace, \oint, start subscript, start color #0d923f, C, start subscript, 1, end subscript, end color #0d923f, end subscript, start color #0c7f99, start bold text, F, end bold text, end color #0c7f99, dot, d, start bold text, r, end bold text, plus, \oint, start subscript, start color #a75a05, C, start subscript, 2, end subscript, end color #a75a05, end subscript, start color #0c7f99, start bold text, F, end bold text, end color #0c7f99, dot, d, start bold text, r, end bold text, end underbrace, start subscript, start text, у, н, и, щ, о, ж, а, в, а, т, space, с, е, space, п, о, space, п, р, о, т, е, ж, е, н, и, е, space, н, а, space, с, е, ч, е, н, и, е, т, о, space, н, а, space, start color #bc2612, S, end color #bc2612, end text, end subscript, equals, \oint, start subscript, start color #bc2612, C, end color #bc2612, end subscript, start color #0c7f99, start bold text, F, end bold text, end color #0c7f99, dot, d, start bold text, r, end bold text

$\displaystyle \underbrace{ \sum_{k = 1}^n \oint_{\redE{C_k}} \blueE{\textbf{F}} \cdot d\textbf{r} }_{\text{Компенсират се по протежение на срязването на $\redE{S}$}} = \oint_{\redE{C}} \blueE{\textbf{F}} \cdot d\textbf{r}$start underbrace, sum, start subscript, k, equals, 1, end subscript, start superscript, n, end superscript, \oint, start subscript, start color #bc2612, C, start subscript, k, end subscript, end color #bc2612, end subscript, start color #0c7f99, start bold text, F, end bold text, end color #0c7f99, dot, d, start bold text, r, end bold text, end underbrace, start subscript, start text, К, о, м, п, е, н, с, и, р, а, т, space, с, е, space, п, о, space, п, р, о, т, е, ж, е, н, и, е, space, н, а, space, с, р, я, з, в, а, н, е, т, о, space, н, а, space, start color #bc2612, S, end color #bc2612, end text, end subscript, equals, \oint, start subscript, start color #bc2612, C, end color #bc2612, end subscript, start color #0c7f99, start bold text, F, end bold text, end color #0c7f99, dot, d, start bold text, r, end bold text

$\displaystyle \underbrace{ \oint_{\redE{C_k}} \blueE{\textbf{F}} \cdot d\textbf{r} }_{\substack{ \text{Интеграл по} \\ \text{по контура на парчето} }} \approx \overbrace{ \Big( ( \text{rot}\, \blueE{\textbf{F}} \goldE{(x_k; y_k; z_k)} ) \cdot \greenE{\hat{\textbf{n}}} \Big) }^{\text{Компонент на ротацията, перпендикулярен на парчето}} \!\!\!\!\!\!\!\!\!\!\!\!\ \underbrace{ \,\redE{d\Sigma} }_{\text{Площ на парчето}}$

$\displaystyle \underbrace{ \oint_{\redE{C_k}} \blueE{\textbf{F}} \cdot d\textbf{r} }_{\substack{ \text{Интеграл по} \\ \text{по контура на парчето} }} \approx \overbrace{ \Big( ( \text{rot}\, \blueE{\textbf{F}} \goldE{(x_k; y_k; z_k)} ) \cdot \greenE{\hat{\textbf{n}}} \Big) }^{\text{Компонент на ротацията, перпендикулярен на парчето}} \!\!\!\!\!\!\!\!\!\!\!\!\ \underbrace{ \,\redE{d\Sigma} }_{\text{Площ на парчето}}$

$\begin{aligned} &\underbrace{ \oint_{\redE{C}} \blueE{\textbf{F}} \cdot d\textbf{r} }_{\substack{ \text{Интеграл по контура}\\ \text{на цялата повърхнина} }} \\\\ &\qquad\qquad \downarrow \\\\ &\underbrace{ \sum_{k = 1}^n \oint_{\redE{C_k}} \blueE{\textbf{F}} \cdot d\textbf{r} }_{\text{Сума от интегралите по контурите на парченцата}} \\\\ &\qquad\qquad \downarrow \\\\ &\underbrace{ \sum_{k = 1}^n \text{rot}\,\blueE{\textbf{F}} \goldE{(x_k; y_k; z_k)} \cdot \greenE{\hat{\textbf{n}}} \;\redE{d\Sigma} }_{\text{Ротационна апроксимация върху всяко парченце}} \end{aligned}$

$\begin{aligned} &\sum_{k = 1}^n \text{rot}\,\blueE{\textbf{F}} \goldE{(x_k; y_k; z_k)} \cdot \greenE{\hat{\textbf{n}}} \;\redE{d\Sigma} \\\\ &\qquad\qquad \downarrow \small{\gray{\text{Когато раздробяваме $\redE{S}$ на все по-малки парченца}}} \\\\ &\iint_{\redE{S}} \text{rot}\,\blueE{\textbf{F}}\cdot\greenE{\hat{\textbf{n}}} \; \redE{d\Sigma} \end{aligned}$

$\begin{aligned} \oint_{\redE{C}} \blueE{\textbf{F}} \cdot d\textbf{r} = \iint_{\redE{S}} \text{rot}\,\blueE{\textbf{F}}\cdot\greenE{\hat{\textbf{n}}} \; \redE{d\Sigma} \end{aligned}$

$\begin{aligned} \iint_{\redE{S}} \text{rot}\,\blueE{\textbf{F}}\cdot\greenE{\hat{\textbf{n}}} \; \redE{d\Sigma} = 0 \end{aligned}$

$\displaystyle \overbrace{ \underbrace{ \iint_\redE{S} }_{\text{$\redE{S}$ е тримерна повърхнина}} \!\!\!\!\!\!\!\!\! \big( \text{rot}\,\blueE{\textbf{F}} \cdot \greenE{\hat{\textbf{n}}} \big) d\Sigma }^{\substack{ \text{Повърхностен интеграл от} \\ \text{ротационно векторно поле} }} = \!\!\!\!\!\!\!\!\! \underbrace{ \int_{\redE{C}} \blueE{\textbf{F}} \cdot d\textbf{r} }_{\substack{ \text{Криволинеен интеграл по} \\ \text{границата на повърхнина} }}$