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Exercises 7.6 Exercises: Scalar Fields and Potentials

Each exercise below repeats the three questions of Example 7.27 in a different physical setting: find the direction of most rapid decrease, the rate along a prescribed direction, and the directions of no change.

1. Electric Potential on a Coated Plate.

The electric potential on a thin coated plate, in volts, is
\begin{equation*} V(x,y) = 100 - 2x^2 - 6y^2, \end{equation*}
with \(x\) and \(y\) in centimeters. Consider the point \(P_0(2,1)\text{.}\)
  1. Find \(\nabla V\) at \(P_0\) and the electric field \(\mathbf E = -\nabla V\) there. Along which unit vector does the potential drop most rapidly, and at what rate?
  2. A test charge is nudged along \(\mathbf u = \tfrac{4}{5}\mathbf i - \tfrac{3}{5}\mathbf j\text{.}\) Compute \(D_{\mathbf u}V\) at \(P_0\text{.}\)
  3. Find the two unit vectors along which the potential does not change, and name the curve they are tangent to.

2. The Fall Line on a Ski Slope.

A mountainside has elevation, in feet,
\begin{equation*} h(x,y) = 3000 - 0.0004x^2 - 0.001y^2, \end{equation*}
with \(x\) and \(y\) in feet. A skier stands at \(P_0(200,100)\text{.}\) Her gravitational potential energy is \(U = mgh\text{,}\) so the downhill force she feels is a positive multiple of \(-\nabla h\text{.}\)
  1. Find the fall line, the unit vector of steepest descent, and the grade of the slope there, expressed as a percent (feet dropped per hundred feet travelled).
  2. A traverse is a route on which she neither climbs nor descends. Find the two unit vectors that accomplish this.
  3. She wants a gentler line of exactly \(12\%\) grade. Using \(D_{\mathbf u}h = \|\nabla h\|\cos\theta\text{,}\) find the angle \(\theta\) between her heading and the fall line.

3. Sailing the Isobars of a Low.

Near the center of a low-pressure system the sea-level pressure, in millibars, is modeled by
\begin{equation*} P(x,y) = 988 + 0.004\left(x^2 + y^2\right), \end{equation*}
with \(x\) and \(y\) in miles from the center of the low. A ship sits at \(P_0(20,10)\text{.}\)
  1. Find the pressure at the ship and \(\nabla P\) there. The pressure-gradient force per unit volume of air is a positive multiple of \(-\nabla P\text{;}\) in which unit direction does it push, and what is \(\|\nabla P\|\) in millibars per mile?
  2. A second ship 40 miles from the center reports a stronger wind. Using this model, explain why \(\|\nabla P\|\) supports that report.
  3. The captain decides to hold a constant barometer reading. Give the two possible unit headings at \(P_0\text{,}\) and describe the shape of the full route.
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