Surface Flux Integrals

Fundamental Theorem for Line Integrals

Divergence/Green’s Theorem

Relates the flow/divergence of a vector field through a surface to the behaviour of the vector field inside the surface, or how much it pierces (enters/exits) the boundary

$$ \oiint_{\partial R} \vec{F} \cdot \hat n dS = \iiint_R \nabla \cdot \vec{F}dV $$

• $S$ is the bounding surface of region $R$ in $R^3$, with $S$ oriented outwards
• $\vec F$ is the vector field at the normal vector for each surface

Divergence Operator

$$ \vec {\nabla} \cdot \vec{F} = \frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y} + \frac{\partial F_z}{\partial z} $$

• Where $\nabla = (\frac{\partial}{\partial x}, \frac{\partial}{\partial y}, \frac{\partial}{\partial z})$ and $F = < F_x, F_y, F_z>$
• Used when you have a three-dimensional region and you want to relate the flux of a vector field through the boundary of the region to the volume integral of the divergence of the vector field over the region
• Use it when you’re dealing with flux (how much of the field passes through) of a vector field out of a closed surface that encloses a volume)

Stoke’s Theorem

Relates the surface integral of the curl of a vector field

$$ \iint \nabla \times \vec{F} \cdot \hat n = \oint _{c = \partial s} \vec F \cdot T ds $$

• Where $\partial s$ is a boundary and $S$ is a surface in $R^3$
• Used when you want to relate the circulation around its boundary to the flux of a curl of a vector field through the surface
• Applied to a surface in 3D
• Used when we’re interested in the circulation around a loop that is the edge of a 3D surface

Curl Operator