Vector Calculus II: Summer-2021 02 : Due 12: M 7/19
Pipe flow line integral
S0 4267S
For \(-3\leq x\leq 3\) feet, the two-dimensional flow rate in feet per second of storm water in a pipe can be modeled by \(6e^{-x^2}\boldsymbol{\hat{y}}\). For each of the following curves, choose a parametrization, construct a line integral of the given vector field over the curve and determine its (approximate) value.
\(x^2+y^2=4\).
\(y=2-x^2\).
\(y=2x\).
Saturn and Titan
S0 4267S
Titan is a moon of Saturn orbiting a distance \(1221.86\times10^3\) kilometers away and whose mass is \(2.36\times10^{-4}\) of Saturn's mass which is \(569\times10^{24}\) kilograms. The potential energy due to gravity is modeled by \(\Phi=\frac{-GMm}{r}\) where \(G=6.67\times10^{-11} \frac{m^3s^2}{kg}\), \(M\) is the mass of Saturn, \(m\) is the mass of an object (such as Titan) and \(r\) is the distance between Saturn and the object in orbit.
Describe the level sets of \(\Phi\) where \(m\) and \(r\) are variables. On which level set is Titan?
Which coordinate system would you choose for calculations in this situation?
Determine the gradient of \(\Phi\). In which direction does it point?