# Calc 3 Week of Nov 28



FTC:

$$
\begin{align}
\int_C \nabla f \cdot d \mathbf{r} 
&=f\left(P_1\right)-f\left(P_0\right) \\
\iint \nabla \times \mathbf F(x,y) \, dA = \iint_D\left(Q_x-P_y\right)\, d A 
&=\int_C \mathbf F\cdot\mathbf T\, ds = \int_C \mathbf{F} \cdot d \mathbf{r} \qquad\text{Green's theorem (Circulation)}\\
\iint\nabla \cdot \mathbf F(x,y) \, dA = \iint_D\left(P_x+Q_y\right) \,d A
&=\int_C \mathbf{F} \cdot \mathbf{N} \,d s  \qquad\text{Green's theorem (Flux)} \\
 \iint_S \nabla \times \mathbf{F} \cdot d \mathbf{S}
 &=\int_C \mathbf{F} \cdot d \mathbf{r} \qquad\text{Stokes theorem} \\
 \iiint_E \nabla \cdot \mathbf{F} \,d V 
 &=\iint_S \mathbf{F} \cdot d \mathbf{S}
\end{align}
$$


- Stokes:
  ![image-20221127182631100](/home/zack/.config/Typora/typora-user-images/image-20221127182631100.png)

  



Some formulas:


$$
\begin{align}
\int_C f(x,y,z)\, ds 
&= \int_a^b f(\mathbf r(t)) \, || \mathbf r'(t)|| \, dt\\
\int_C \mathbf F(x,y,z)\cdot d\mathbf r 
&= \int_a^b\mathbf F(\mathbf r(t))\cdot \mathbf{r}'(t)\,dt \\
\iint_S f(x,y,z) \, dS  
&= \iint f(\mathbf r(u, v)) \, || \mathbf r_u(u, v) \times \mathbf r_v(u, v)|| \, dA, \\
\iint_S \mathbf F(x,y,z)\cdot d\mathbf S 
&= \iint_S \mathbf F(\mathbf r(u, v))\cdot \mathbf r'(u, v)   \, dA \\
\iint_S \mathbf{F}\cdot \mathbf{N} \, dS 
&= \iint \mathbf F(\mathbf r(u, v)) \cdot (\mathbf r_u \times \mathbf r_v) \, dA \\ \\
S = g(x,y,z) \implies \mathbf r(u, v)
&= (u, v, g(u, v)) \implies ||\mathbf r_u \times \mathbf r_v|| = \sqrt{1 + g_u^2 + g_v^2} \\
 \mu(S) = \iint 1\,dS
\end{align}
$$