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# Divergence, Curl, Surface Integrals, Stokes' Theorem, Divergence Theorem ## Divergence and Curl * **Divergence:** Measures the "outflowing-ness" of a vector field at a point. * For a dot product: $\text{div } \vec{F} = \frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y} + \frac...

# Divergence, Curl, Surface Integrals, Stokes' Theorem, Divergence Theorem ## Divergence and Curl * **Divergence:** Measures the "outflowing-ness" of a vector field at a point. * For a dot product: $\text{div } \vec{F} = \frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y} + \frac{\partial F_z}{\partial z}$ * $\vec{F} = 0$ source-free field * **Curl:** Measures the "rotation" or "twisting" of a vector field. * Curl $\vec{F} = \vec{\nabla} \times \vec{F}$ * $\vec{F} = 0$ irrotational field ## Surface Integrals * **Scalar field $f(x, y, z)$:** $\iint_S f(x, y, z) \,dS = \iint_D f(r(u,v)) \|\vec{r}_u \times \vec{r}_v\| \,dA$ * **Vector field $\vec{F}$:** $\iint_S \vec{F} \cdot \vec{n} \,dS = \iint_D \vec{F}(r(u,v)) \cdot (\vec{r}_u \times \vec{r}_v) \,dA$ * Where $n$ is the unit normal vector to S. * Flux is the total amount of a vector field $\vec{F}$ passing through a surface $S$. ## Stokes' Theorem * Relates a surface integral of curl $\vec{F}$ to a line integral around the boundary curve of $S$: $\iint_S (\vec{\nabla} \times \vec{F}) \cdot \vec{n} \,dS = \oint_C \vec{F} \cdot d\vec{r}$ * Where: * $C$ is the boundary curve of $S$. * $dS$ is the differential area element. * The conditions: * Surface is oriented and smooth. * Boundary curve is simple, closed, and smooth with a positive orientation. * Vector field has continuous partial derivatives. ## Divergence Theorem * Relates the flux of a vector field through a closed surface $S$ to a triple integral over the volume $V$ enclosed by $S$: $\iint_S \vec{F} \cdot \vec{n} \, dS = \iiint_V \vec{\nabla} \cdot \vec{F} \, dV$ * Use for computing net outward flux and simplifying flux integrals for symmetric regions.