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Научный форум dxdyМатематика, Физика, Computer Science, Machine Learning, LaTeX, Механика и Техника, Химия,Биология и Медицина, Экономика и Финансовая Математика, Гуманитарные науки |
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Научный форум dxdyМатематика, Физика, Computer Science, Machine Learning, LaTeX, Механика и Техника, Химия,Биология и Медицина, Экономика и Финансовая Математика, Гуманитарные науки |
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| Страница 1 из 2 |
[ Сообщений: 30 ] | На страницу 1, 2 След. |
![\[\frac{{\partial f(x,y)}}{{\partial x}} = {g_1}(x,y)\]](https://dxdy.ru/math/7b603cc95df694c9519cc8baf02bba1b82.png "\[\frac{{\partial f(x,y)}}{{\partial x}} = {g_1}(x,y)\]")
![\[\frac{{\partial f(x,y)}}{{\partial y}} = {g_2}(x,y)\]](https://dxdy.ru/math/db679ae3aac2c9fd735cd7413038d61b82.png "\[\frac{{\partial f(x,y)}}{{\partial y}} = {g_2}(x,y)\]")
![\[\frac{{\partial f}}{{\partial x}} = \frac{{ (x - a)}}{{\left( {{{(x - a)}^2} + {y^2}} \right)}}\]](https://dxdy.ru/math/d54dc87091ecd923bd43554a9c793cb682.png "\[\frac{{\partial f}}{{\partial x}} = \frac{{ (x - a)}}{{\left( {{{(x - a)}^2} + {y^2}} \right)}}\]")
![\[\frac{{\partial f}}{{\partial y}} = \frac{{ y}}{{\left( {{{(x - a)}^2} + {y^2}} \right)}}\]](https://dxdy.ru/math/e3bb5e03eed3e66bcf5102c4232c770e82.png "\[\frac{{\partial f}}{{\partial y}} = \frac{{ y}}{{\left( {{{(x - a)}^2} + {y^2}} \right)}}\]")
![\[f = \frac{1}{2}\ln \left( {{{(x - a)}^2} + {y^2}} \right) + C\]](https://dxdy.ru/math/413ed489146458f4928f85f3df4eaf6782.png "\[f = \frac{1}{2}\ln \left( {{{(x - a)}^2} + {y^2}} \right) + C\]")
![\[{\rm d}f=\frac{{(x - a)}}{{\left( {{{(x - a)}^2} + {y^2}} \right)}}dx + \frac{y}{{\left( {{{(x - a)}^2} + {y^2}} \right)}}dy\]](https://dxdy.ru/math/f34bb80d29b1c584d142b0a1ad23c0c082.png "\[{\rm d}f=\frac{{(x - a)}}{{\left( {{{(x - a)}^2} + {y^2}} \right)}}dx + \frac{y}{{\left( {{{(x - a)}^2} + {y^2}} \right)}}dy\]")
![\[\int\limits_{({x_0},{y_0})}^{(x,y)} {\frac{{(x - a)}}{{\left( {{{(x - a)}^2} + {y^2}} \right)}}dx + \frac{y}{{\left( {{{(x - a)}^2} + {y^2}} \right)}}dy} = \]](https://dxdy.ru/math/cd77704d906e42deb128a46c677e7e9282.png "\[\int\limits_{({x_0},{y_0})}^{(x,y)} {\frac{{(x - a)}}{{\left( {{{(x - a)}^2} + {y^2}} \right)}}dx + \frac{y}{{\left( {{{(x - a)}^2} + {y^2}} \right)}}dy} = \]")
![\[ = \int\limits_{{x_0}}^x {\frac{{(x - a)}}{{\left( {{{(x - a)}^2} + y_0^2} \right)}}dx} + \int\limits_{{y_0}}^y {\frac{y}{{\left( {{{(x - a)}^2} + {y^2}} \right)}}dy} = \]](https://dxdy.ru/math/b93a8d95c2d6971cd3624485b2ea572882.png "\[ = \int\limits_{{x_0}}^x {\frac{{(x - a)}}{{\left( {{{(x - a)}^2} + y_0^2} \right)}}dx} + \int\limits_{{y_0}}^y {\frac{y}{{\left( {{{(x - a)}^2} + {y^2}} \right)}}dy} = \]")
![\[ = \frac{1}{2}\ln \left( {{{(x - a)}^2} + y_0^2} \right) - \frac{1}{2}\ln \left( {{{({x_0} - a)}^2} + y_0^2} \right) + \frac{1}{2}\ln \left( {{{(x - a)}^2} + {y^2}} \right) - \frac{1}{2}\ln \left( {{{(x - a)}^2} + y_0^2} \right) = \]](https://dxdy.ru/math/ceb00253d56838eaf938249def4e8fe382.png "\[ = \frac{1}{2}\ln \left( {{{(x - a)}^2} + y_0^2} \right) - \frac{1}{2}\ln \left( {{{({x_0} - a)}^2} + y_0^2} \right) + \frac{1}{2}\ln \left( {{{(x - a)}^2} + {y^2}} \right) - \frac{1}{2}\ln \left( {{{(x - a)}^2} + y_0^2} \right) = \]")
![\[ = \frac{1}{2}\ln \left( {{{(x - a)}^2} + {y^2}} \right) - \frac{1}{2}\ln \left( {{{({x_0} - a)}^2} + y_0^2} \right).\]](https://dxdy.ru/math/f1d68767634f558c46b2d6c0d33b361a82.png "\[ = \frac{1}{2}\ln \left( {{{(x - a)}^2} + {y^2}} \right) - \frac{1}{2}\ln \left( {{{({x_0} - a)}^2} + y_0^2} \right).\]")