გრინის თეორემა

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

$\displaystyle \oint_\greenE{C_1} \blueE{F} \cdot d\textbf{r} + \oint_\goldE{C_2} \blueE{F} \cdot d\textbf{r}$\oint, start subscript, start color #0d923f, C, start subscript, 1, end subscript, end color #0d923f, end subscript, start color #0c7f99, F, 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, F, end color #0c7f99, dot, d, start bold text, r, end bold text

$\displaystyle \oint_\greenE{C_1} \blueE{F} \cdot d\textbf{r} + \oint_\goldE{C_2} \blueE{F} \cdot d\textbf{r} = \oint_\redE{C} \blueE{F} \cdot d\textbf{r}$\oint, start subscript, start color #0d923f, C, start subscript, 1, end subscript, end color #0d923f, end subscript, start color #0c7f99, F, 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, F, end color #0c7f99, dot, d, start bold text, r, end bold text, equals, \oint, start subscript, start color #bc2612, C, end color #bc2612, end subscript, start color #0c7f99, F, end color #0c7f99, dot, d, start bold text, r, end bold text

$\displaystyle \oint_\redE{C_1} \blueE{F} \cdot d\textbf{r} + \oint_\redE{C_2} \blueE{F} \cdot d\textbf{r} + \oint_\redE{C_3} \blueE{F} \cdot d\textbf{r} + \oint_\redE{C_4} \blueE{F} \cdot d\textbf{r} = \oint_\redE{C} \blueE{F} \cdot d\textbf{r}$\oint, start subscript, start color #bc2612, C, start subscript, 1, end subscript, end color #bc2612, end subscript, start color #0c7f99, F, end color #0c7f99, dot, d, start bold text, r, end bold text, plus, \oint, start subscript, start color #bc2612, C, start subscript, 2, end subscript, end color #bc2612, end subscript, start color #0c7f99, F, end color #0c7f99, dot, d, start bold text, r, end bold text, plus, \oint, start subscript, start color #bc2612, C, start subscript, 3, end subscript, end color #bc2612, end subscript, start color #0c7f99, F, end color #0c7f99, dot, d, start bold text, r, end bold text, plus, \oint, start subscript, start color #bc2612, C, start subscript, 4, end subscript, end color #bc2612, end subscript, start color #0c7f99, F, end color #0c7f99, dot, d, start bold text, r, end bold text, equals, \oint, start subscript, start color #bc2612, C, end color #bc2612, end subscript, start color #0c7f99, F, end color #0c7f99, dot, d, start bold text, r, end bold text

$\displaystyle \sum_{k = 1}^n \left( \oint_\redE{C_k} \blueE{F} \cdot d\textbf{r} \right) = \oint_\redE{C} \blueE{F} \cdot d\textbf{r}$sum, start subscript, k, equals, 1, end subscript, start superscript, n, end superscript, left parenthesis, \oint, start subscript, start color #bc2612, C, start subscript, k, end subscript, end color #bc2612, end subscript, start color #0c7f99, F, end color #0c7f99, dot, d, start bold text, r, end bold text, right parenthesis, equals, \oint, start subscript, start color #bc2612, C, end color #bc2612, end subscript, start color #0c7f99, F, 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{ინტეგრალი a-ს გარშემო} \\ \text{მცირე ნაწილი $\redE{R_k}$} }} \approx \left( \text{2d-როტორი}\,\blueE{\textbf{F}} \underbrace{ \goldE{(x_k, y_k)} }_{\text{Point in $\redE{R_k}$}} \right) \underbrace{|\redE{R_k}|}_{\text{$\redE{R_k}$-ის ფართობი}}$

$\displaystyle \sum_{k = 1}^n \left( \oint_\redE{C_k} \blueE{\textbf{F}} \cdot d\textbf{r} \right) \approx \sum_{k = 1}^n \left( \text{2d-როტორი}\,\blueE{\textbf{F}} \underbrace{ \goldE{(x_k, y_k)} }_{\text{წერტილი $\redE{R_k}$-ში}} \,|\redE{R_k}| \right)$sum, start subscript, k, equals, 1, end subscript, start superscript, n, end superscript, left parenthesis, \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, right parenthesis, approximately equals, sum, start subscript, k, equals, 1, end subscript, start superscript, n, end superscript, left parenthesis, start text, 2, d, negative, რ, ო, ტ, ო, რ, ი, end text, start color #0c7f99, start bold text, F, end bold text, end color #0c7f99, start underbrace, start color #a75a05, left parenthesis, x, start subscript, k, end subscript, comma, y, start subscript, k, end subscript, right parenthesis, end color #a75a05, end underbrace, start subscript, start text, წ, ე, რ, ტ, ი, ლ, ი, space, start color #bc2612, R, start subscript, k, end subscript, end color #bc2612, negative, შ, ი, end text, end subscript, vertical bar, start color #bc2612, R, start subscript, k, end subscript, end color #bc2612, vertical bar, right parenthesis

$\displaystyle \oint_\redE{C} \blueE{\textbf{F}} \cdot d\textbf{r} \approx \sum_{k = 1}^n \left( \text{2d-როტორი}\,\blueE{\textbf{F}} \underbrace{ \goldE{(x_k, y_k)} }_{\text{წერტილი $\redE{R_k}$-ში}} \,|\redE{R_k}| \right)$\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, approximately equals, sum, start subscript, k, equals, 1, end subscript, start superscript, n, end superscript, left parenthesis, start text, 2, d, negative, რ, ო, ტ, ო, რ, ი, end text, start color #0c7f99, start bold text, F, end bold text, end color #0c7f99, start underbrace, start color #a75a05, left parenthesis, x, start subscript, k, end subscript, comma, y, start subscript, k, end subscript, right parenthesis, end color #a75a05, end underbrace, start subscript, start text, წ, ე, რ, ტ, ი, ლ, ი, space, start color #bc2612, R, start subscript, k, end subscript, end color #bc2612, negative, შ, ი, end text, end subscript, vertical bar, start color #bc2612, R, start subscript, k, end subscript, end color #bc2612, vertical bar, right parenthesis

$\displaystyle \sum_{k = 1}^n \left( \text{2d-როტორი}\,\blueE{\textbf{F}} \underbrace{ \goldE{(x_k, y_k)} }_{\text{წერტილი $\redE{R_k}$-ში}} \,|\redE{R_k}| \right) \rightarrow \iint_\redE{R} \text{2d-როტორი}\,\blueE{\textbf{F}} \,\redE{dA}$sum, start subscript, k, equals, 1, end subscript, start superscript, n, end superscript, left parenthesis, start text, 2, d, negative, რ, ო, ტ, ო, რ, ი, end text, start color #0c7f99, start bold text, F, end bold text, end color #0c7f99, start underbrace, start color #a75a05, left parenthesis, x, start subscript, k, end subscript, comma, y, start subscript, k, end subscript, right parenthesis, end color #a75a05, end underbrace, start subscript, start text, წ, ე, რ, ტ, ი, ლ, ი, space, start color #bc2612, R, start subscript, k, end subscript, end color #bc2612, negative, შ, ი, end text, end subscript, vertical bar, start color #bc2612, R, start subscript, k, end subscript, end color #bc2612, vertical bar, right parenthesis, right arrow, \iint, start subscript, start color #bc2612, R, end color #bc2612, end subscript, start text, 2, d, negative, რ, ო, ტ, ო, რ, ი, end text, start color #0c7f99, start bold text, F, end bold text, end color #0c7f99, start color #bc2612, d, A, end color #bc2612

$\begin{aligned} \oint_\redE{C} \blueE{\textbf{F}} \cdot d\textbf{r} &= \sum_{k = 1}^n \left( \oint_\redE{C_k} \blueE{\textbf{F}} \cdot d\textbf{r} \right) \\\\ &\approx \sum_{k = 1}^n \left( \text{2d-როტორი}\,\blueE{\textbf{F}} \underbrace{ \goldE{(x_k, y_k)} }_{\text{წერტილი $\redE{R_k}$-ში}} \,|\redE{R_k}| \right) \\\\ &\to \iint_\redE{R} \text{2d-როტორი}\,\blueE{\textbf{F}} \,\redE{dA} \end{aligned}$

$\large \displaystyle \oint_\redE{C} \blueE{\textbf{F}} \cdot d\textbf{r} = \iint_\redE{R} \text{2d-როტორი}\,\blueE{\textbf{F}} \,\redE{dA}$\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, equals, \iint, start subscript, start color #bc2612, R, end color #bc2612, end subscript, start text, 2, d, negative, რ, ო, ტ, ო, რ, ი, end text, start color #0c7f99, start bold text, F, end bold text, end color #0c7f99, start color #bc2612, d, A, end color #bc2612

$\displaystyle \oint_\redE{C} P\,dx + Q\,dy = \iint_\redE{R} \left( \dfrac{\partial Q}{\partial x} - \dfrac{\partial P}{\partial y} \right) \, dA$\oint, start subscript, start color #bc2612, C, end color #bc2612, end subscript, P, d, x, plus, Q, d, y, equals, \iint, start subscript, start color #bc2612, R, end color #bc2612, end subscript, left parenthesis, start fraction, \partial, Q, divided by, \partial, x, end fraction, minus, start fraction, \partial, P, divided by, \partial, y, end fraction, right parenthesis, d, A

$\displaystyle \blueE{\textbf{F}}(x, y) = P(x, y)\hat{\textbf{i}} + Q(x, y)\hat{\textbf{j}} = \left[ \begin{array}{c} P(x, y) \\ Q(x, y) \end{array} \right]$

$\displaystyle \underbrace{ \oint_\redE{C_k} \blueE{\textbf{F}} \cdot d\textbf{r} }_{\substack{ \text{ინტეგრალი a-ს გარშემო} \\ \text{მცირე ნაწილი $\redE{R_k}$} }} \approx \left( \text{2d-როტორი}\,\blueE{\textbf{F}} \underbrace{ \goldE{(x_k, y_k)} }_{\text{Point in $\redE{R_k}$}} \right) \underbrace{|\redE{R_k}|}_{\text{$\redE{R_k}$-ის ფართობი}}$

$\displaystyle \oint_\redE{C} \blueE{\textbf{F}} \cdot d\textbf{r} = \iint_\redE{R} \text{2d-როტორი}\,\blueE{\textbf{F}} \,\redE{dA}$\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, equals, \iint, start subscript, start color #bc2612, R, end color #bc2612, end subscript, start text, 2, d, negative, რ, ო, ტ, ო, რ, ი, end text, start color #0c7f99, start bold text, F, end bold text, end color #0c7f99, start color #bc2612, d, A, end color #bc2612