Stokes teoremi

$\displaystyle\oint_\redE{C}\blueE{\textbf{F}} \cdot d \textbf{r}$\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

$\displaystyle \underbrace{ \oint_{\greenE{C_1}} \blueE{\textbf{F}} \cdot d\textbf{r} + \oint_{\goldE{C_2}} \blueE{\textbf{F}} \cdot d\textbf{r} }_{\text{$\redE{S}$ boyunca dilimi yok edin}} = \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, start color #bc2612, S, end color #bc2612, space, b, o, y, u, n, c, a, space, d, i, l, i, m, i, space, y, o, k, space, e, d, i, n, 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{Cancel out along slice through $\redE{S}$}} = \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, C, a, n, c, e, l, space, o, u, t, space, a, l, o, n, g, space, s, l, i, c, e, space, t, h, r, o, u, g, h, 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

${\displaystyle \underset{\begin{array}{c}\text{Parçanın sınırının }\\ \text{etrafındaki integral}\end{array}}{\underbrace{{\oint }_{C_k}\mathbf{\text{F}}\cdot d\mathbf{\text{r}}}}\approx \stackrel{\text{Parçaya dik olan rotasyonelin bileşeni}}{\overbrace{((\text{curl}\mathbf{\text{F}}(x_k,y_k,z_k))\cdot \widehat{\mathbf{\text{n}}})}}\underset{\text{Parçanın alanı}}{\underbrace{d\mathrm{\Sigma }}}}$ ‍

${\displaystyle \underset{\begin{array}{c}\text{Parçanın sınırının }\\ \text{etrafındaki integral}\end{array}}{\underbrace{{\oint }_{C_k}\mathbf{\text{F}}\cdot d\mathbf{\text{r}}}}\approx \stackrel{\text{Parçaya dik olan rotasyonelin bileşeni}}{\overbrace{((\text{curl}\mathbf{\text{F}}(x_k,y_k,z_k))\cdot \widehat{\mathbf{\text{n}}})}}\underset{\text{Parçanın alanı}}{\underbrace{d\mathrm{\Sigma }}}}$ ‍

$\begin{aligned} &\underbrace{ \oint_{\redE{C}} \blueE{\textbf{F}} \cdot d\textbf{r} }_{\substack{ \text{Tüm yüzeyin sınırının}\\ \text{etrafındaki integral} }} \\\\ &\qquad\qquad \downarrow \\\\ &\underbrace{ \sum_{k = 1}^n \oint_{\redE{C_k}} \blueE{\textbf{F}} \cdot d\textbf{r} }_{\text{Küçük parçacıkların etrafındaki integrallerin toplamı}} \\\\ &\qquad\qquad \downarrow \\\\ &\underbrace{ \sum_{k = 1}^n \text{rotasyonel}\,\blueE{\textbf{F}} \goldE{(x_k, y_k, z_k)} \cdot \greenE{\hat{\textbf{n}}} \;\redE{d\Sigma} }_{\text{Her parçaya rotasyonel kestirimi uygulayın}} \end{aligned}$

$\begin{aligned} &\sum_{k = 1}^n \text{curl}\,\blueE{\textbf{F}} \goldE{(x_k, y_k, z_k)} \cdot \greenE{\hat{\textbf{n}}} \;\redE{d\Sigma} \\\\ &\qquad\qquad \downarrow \small{\gray{\text{As $\redE{S}$ daha ince ve daha çok kesilmiştir}}} \\\\ &\iint_{\redE{S}} \text{curl}\,\blueE{\textbf{F}}\cdot\greenE{\hat{\textbf{n}}} \; \redE{d\Sigma} \end{aligned}$

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

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

$\displaystyle \overbrace{ \underbrace{ \iint_\redE{S} }_{\text{$\redE{S}$ 3 boyutlu bir yüzeydir}} \!\!\!\!\!\!\!\!\! \big( \text{curl}\,\blueE{\textbf{F}} \cdot \greenE{\hat{\textbf{n}}} \big) d\Sigma }^{\substack{ \text{Bir rotasyonel vektör alanının} \\ \text{yüzey integrali} }} = \!\!\!\!\!\!\!\!\! \underbrace{ \int_{\redE{C}} \blueE{\textbf{F}} \cdot d\textbf{r} }_{\substack{ \text{Yüzeyin sınırı etrafındaki} \\ \text{çizgi integrali} }}$