assignment Homework
Find the upward pointing flux of the electric field \(\vec E =E_0\, z\, \hat z\) through the part of the surface \(z=-3 s^2 +12\) (cylindrical coordinates) that sits above the \((x, y)\)--plane.
group Small Group Activity
30 min.
assignment Homework
Consider the vector field \(\vec F=(x+2)\hat{x} +(z+2)\hat{z}\).
assignment Homework
The gravitational field due to a spherical shell of matter (or equivalently, the electric field due to a spherical shell of charge) is given by: \begin{equation} \vec g = \begin{cases} 0&\textrm{for } r<a\\ -G \,\frac{M}{b^3-a^3}\, \left( r-\frac{a^3}{r^2}\right)\, \hat r & \textrm{for } a<r<b\\ -G\,\frac{M}{r^2}\, \hat r & \textrm{for } r>b \\ \end{cases} \end{equation}
This problem explores the consequences of the divergence theorem for this shell.
assignment Homework
assignment Homework
Shown above is a two-dimensional cross-section of a vector field. All the parallel cross-sections of this field look exactly the same. Determine the direction of the curl at points A, B, and C.
assignment Homework
assignment Homework
Calculate the divergence of each of the following vector fields. You may look up the formulas for divergence in curvilinear coordinates.
group Small Group Activity
30 min.
Students use known algebraic expressions for vector line elements \(d\vec{r}\) to determine all simple vector area \(d\vec{A}\) and volume elements \(d\tau\) in cylindrical and spherical coordinates.
This activity is identical to Scalar Surface and Volume Elements except uses a vector approach to find directed surface and volume elements.
group Small Group Activity
5 min.
Shown above is a two-dimensional vector field.
Determine whether the divergence at point A and at point C is positive, negative, or zero.