Activity: lower anchor MTH 255 (editing)

$d \vec r$ represents a small displacement vector (i.e., it points along the direction of a step with a magnitude equal to the length of that step).

In each coordinate system, there exists a formula for the divergance. The standard examples are: \[\begin{align}\vec\nabla\cdot\vec v &= \frac{\partial v_x}{\partial x} + \frac{\partial v_y}{\partial y}+\frac{\partial v_z}{\partial z} \text{ in rectangular coodinates} \\ \vec\nabla\cdot\vec v &= \frac{1}{s}\frac{\partial \left(s v_s\right)}{\partial s} + \frac{1}{s}\frac{\partial v_\phi}{\partial \phi}+\frac{\partial v_z}{\partial z} \text{ in cylindrical coodinates} \\ \vec\nabla\cdot\vec v &= \frac{1}{r^2}\frac{\partial \left(r^2 v_r\right)}{\partial r} + \frac{1}{r \sin(\theta)}\frac{\partial \left(\sin(\theta)\ v_\theta\right)}{\partial \theta}+\frac{1}{r \sin(\theta)}\frac{\partial v_\phi}{\partial \phi} \text{ in spherical coodinates}\end{align}\]

In any coordinate system, there exists a formula for the curl. Examples include:\[\vec\nabla\times\vec v = \left(\frac{\partial v_z}{\partial y}-\frac{\partial v_y}{\partial z}\right)\hat x + \left(\frac{\partial v_x}{\partial z}-\frac{\partial v_z}{\partial x}\right)\hat y + \left(\frac{\partial v_y}{\partial x}-\frac{\partial v_x}{\partial y}\right)\hat z \text{ in rectangular coodinates}\]\[\vec\nabla\times\vec v = \left(\frac{1}{s}\frac{\partial v_z}{\partial \phi} - \frac{\partial v_\phi}{\partial z}\right) \hat s + \left(\frac{\partial v_s}{\partial z} - \frac{\partial v_z}{\partial s}\right) \hat \phi + \frac{1}{s}\left(\frac{\partial \left(s\ v_\phi\right)}{\partial s} - \frac{\partial v_s}{\partial \phi}\right) \hat z \text{ in cylindrical coodinates}\]\[\vec\nabla\times\vec v = \frac{1}{r \sin(\theta)}\left(\frac{\partial \left(\sin(\theta)\ v_\phi\right)}{\partial \theta} - \frac{\partial v_\theta}{\partial \phi}\right) \hat r + \frac{1}{r}\left(\frac{1}{\sin(\theta)}\frac{\partial v_r}{\partial \phi} - \frac{\partial \left(r\ v_\phi\right)}{\partial r}\right)\hat \theta +\frac{1}{r} \left(\frac{\partial \left(r\ v_\theta\right)}{\partial r} - \frac{\partial v_r}{\partial \theta}\right) \hat \phi \text{ in spherical coodinates}\]

1. The total divergence in a volume is the same as the total flux out of its surface.[]2) The divergence theorem is used to move between the integral and differential forms of Gauss's law.[]3) Dimensions are the same

FIXME