# Calc III Lectures Goal: Section 6.4, define flux at the beginning ## Background Arc length: ![image-20221106160531673](/home/zack/.config/Typora/typora-user-images/image-20221106160531673.png) ![image-20221106160539213](/home/zack/.config/Typora/typora-user-images/image-20221106160539213.png) ![image-20221106160626168](/home/zack/.config/Typora/typora-user-images/image-20221106160626168.png) Note that $$\int_{t_0}^{t_1} \left|| \vec r'(t)\right|| \, dt = \displaystyle\int_{t_0}^{t_1} \sqrt{ \displaystyle\sum_{i\geq 0} x_i'(t)^2 } \, dt$$, so $s = \vec r(t) \implies ds = \left|| \vec r'(t) \right|| \, dt$ and ![image-20221106160823886](/home/zack/.config/Typora/typora-user-images/image-20221106160823886.png) Here $\bf T$ is the normalized unit tangent vector ![image-20221106161123563](/home/zack/.config/Typora/typora-user-images/image-20221106161123563.png) ![image-20221106161203947](/home/zack/.config/Typora/typora-user-images/image-20221106161203947.png) ![image-20221106161358121](/home/zack/.config/Typora/typora-user-images/image-20221106161358121.png) ## Flux ![image-20221106162431551](/home/zack/.config/Typora/typora-user-images/image-20221106162431551.png) ![image-20221106162501939](/home/zack/.config/Typora/typora-user-images/image-20221106162501939.png) ![image-20221106162809926](/home/zack/.config/Typora/typora-user-images/image-20221106162809926.png) Black holes: $10^{20}\, \mathrm{kg}$. Circulation: measures tendency of a fluid to move along a specific path $C$: ![image-20221106163429081](/home/zack/.config/Typora/typora-user-images/image-20221106163429081.png) ![image-20221106163445291](/home/zack/.config/Typora/typora-user-images/image-20221106163445291.png) ## Green's Theorem Idea: extend FTC to higher dimensions. - Dim 1: $\int_a^b F^{\prime}(x) d x=F(b)-F(a)$ ![image-20221106164306650](/home/zack/.config/Typora/typora-user-images/image-20221106164306650.png) - Line integrals: $\int_C \nabla f \cdot d \mathbf{r}=f(\mathbf{r}(b))-f(\mathbf{r}(a))$ Pic of curve with $\partial C = \{p_0, p_1\}$. - Double integrals: Green's theorem. - Triple integrals: Stokes' theorem - Higher dimensions and manifolds: $\int_{\partial \Omega} \omega=\int_{\Omega} d \omega$ where $dF = \nabla F, \nabla \times(F), \nabla\cdot F$,etc. Generally $A_3 \xrightarrow{\nabla\cdot} A_2 \xrightarrow{\nabla\times} A_1 \xrightarrow{\nabla} \bf{R}^3$, and $$\nabla \cdot(\nabla \times F) = 0, \nabla\times(\nabla F) = 0$$ (divergence of curl is zero, curl of gradient is zero). - TODO: find a cool example of using this! ![image-20221106173810259](/home/zack/.config/Typora/typora-user-images/image-20221106173810259.png) ## Circulation Form ![image-20221106165432160](/home/zack/.config/Typora/typora-user-images/image-20221106165432160.png) ![image-20221106165500699](/home/zack/.config/Typora/typora-user-images/image-20221106165500699.png) ![image-20221106165532123](/home/zack/.config/Typora/typora-user-images/image-20221106165532123.png) ### Example ![image-20221106165842092](/home/zack/.config/Typora/typora-user-images/image-20221106165842092.png) ![image-20221106165903236](/home/zack/.config/Typora/typora-user-images/image-20221106165903236.png) ### Example ![image-20221106165927949](/home/zack/.config/Typora/typora-user-images/image-20221106165927949.png) ![image-20221106165940828](/home/zack/.config/Typora/typora-user-images/image-20221106165940828.png) ## Example ![image-20221106170554075](/home/zack/.config/Typora/typora-user-images/image-20221106170554075.png) ![image-20221106170612544](/home/zack/.config/Typora/typora-user-images/image-20221106170612544.png) ## Comment on areas ![image-20221106170707134](/home/zack/.config/Typora/typora-user-images/image-20221106170707134.png) ## Flux ![image-20221106170751264](/home/zack/.config/Typora/typora-user-images/image-20221106170751264.png) ## Example ![image-20221106170820838](/home/zack/.config/Typora/typora-user-images/image-20221106170820838.png) ![image-20221106170830960](/home/zack/.config/Typora/typora-user-images/image-20221106170830960.png) ## Example ![image-20221106170932792](/home/zack/.config/Typora/typora-user-images/image-20221106170932792.png) ![image-20221106170943309](/home/zack/.config/Typora/typora-user-images/image-20221106170943309.png) ## Example ![image-20221106171000264](/home/zack/.config/Typora/typora-user-images/image-20221106171000264.png) ![image-20221106171007882](/home/zack/.config/Typora/typora-user-images/image-20221106171007882.png) ## Source-free fields ![image-20221106171101001](/home/zack/.config/Typora/typora-user-images/image-20221106171101001.png) ## Example ![image-20221106171153777](/home/zack/.config/Typora/typora-user-images/image-20221106171153777.png) ![image-20221106171201143](/home/zack/.config/Typora/typora-user-images/image-20221106171201143.png) ## Harmonic Functions ![image-20221106171501145](/home/zack/.config/Typora/typora-user-images/image-20221106171501145.png) ## Regions with holes ![image-20221106171605839](/home/zack/.config/Typora/typora-user-images/image-20221106171605839.png) ### Example ![image-20221106171652602](/home/zack/.config/Typora/typora-user-images/image-20221106171652602.png) # Remarks on generalizations ![image-20221106172349722](/home/zack/.config/Typora/typora-user-images/image-20221106172349722.png) ![image-20221106172436449](/home/zack/.config/Typora/typora-user-images/image-20221106172436449.png) ![image-20221106172529340](/home/zack/.config/Typora/typora-user-images/image-20221106172529340.png)