Directional Derivative

    • group DELETE Navigating a Hill

      group Small Group Activity

      30 min.

      DELETE Navigating a Hill
      Static Fields 2022 (4 years) In this small group activity, students determine various aspects of local points on an elliptic hill which is a function of two variables. The gradient is emphasized as a local quantity which points in the direction of greatest change at a point in the scalar field.
    • group Earthquake waves

      group Small Group Activity

      30 min.

      Earthquake waves
      Contemporary Challenges 2022 (4 years)

      wave equation speed of sound

      In this activity students use the known speed of earthquake waves to estimate the Young's modulus of the Earth's crust.
    • face Time Evolution Refresher (Mini-Lecture)

      face Lecture

      30 min.

      Time Evolution Refresher (Mini-Lecture)
      Central Forces 2023 (3 years)

      schrodinger equation time dependence stationary states

      Quantum Ring Sequence

      The instructor gives a brief lecture about time dependence of energy eigenstates (e.g. McIntyre, 3.1). Notes for the students are attached.
    • assignment Mass-radius relationship for white dwarfs

      assignment Homework

      Mass-radius relationship for white dwarfs
      White dwarf Mass Density Energy Thermal and Statistical Physics 2020

      Consider a white dwarf of mass \(M\) and radius \(R\). The dwarf consists of ionized hydrogen, thus a bunch of free electrons and protons, each of which are fermions. Let the electrons be degenerate but nonrelativistic; the protons are nondegenerate.

      1. Show that the order of magnitude of the gravitational self-energy is \(-\frac{GM^2}{R}\), where \(G\) is the gravitational constant. (If the mass density is constant within the sphere of radius \(R\), the exact potential energy is \(-\frac53\frac{GM^2}{R}\)).

      2. Show that the order of magnitude of the kinetic energy of the electrons in the ground state is \begin{align} \frac{\hbar^2N^{\frac53}}{mR^2} \approx \frac{\hbar^2M^{\frac53}}{mM_H^{\frac53}R^2} \end{align} where \(m\) is the mass of an electron and \(M_H\) is the mas of a proton.

      3. Show that if the gravitational and kinetic energies are of the same order of magnitude (as required by the virial theorem of mechanics), \(M^{\frac13}R \approx 10^{20} \text{g}^{\frac13}\text{cm}\).

      4. If the mass is equal to that of the Sun (\(2\times 10^{33}g\)), what is the density of the white dwarf?

      5. It is believed that pulsars are stars composed of a cold degenerate gas of neutrons (i.e. neutron stars). Show that for a neutron star \(M^{\frac13}R \approx 10^{17}\text{g}^{\frac13}\text{cm}\). What is the value of the radius for a neutron star with a mass equal to that of the Sun? Express the result in \(\text{km}\).

    • group Optical depth of atmosphere

      group Small Group Activity

      30 min.

      Optical depth of atmosphere
      Contemporary Challenges 2022 (4 years) In this activity students estimate the optical depth of the atmosphere at the infrared wavelength where carbon dioxide has peak absorption.
    • group The Hill

      group Small Group Activity

      30 min.

      The Hill
      Vector Calculus II 2022 (4 years)

      Gradient

      Gradient Sequence

      In this small group activity, students determine various aspects of local points on an elliptic hill which is a function of two variables. The gradient is emphasized as a local quantity which points in the direction of greatest change at a point in the scalar field.
    • assignment Phase 2

      assignment Homework

      Phase 2
      quantum mechanics relative phase overall phase measurement probability Quantum Fundamentals 2022 (2 years) Consider the three quantum states: \[\left\vert \psi_1\right\rangle = \frac{4}{5}\left\vert +\right\rangle+ i\frac{3}{5} \left\vert -\right\rangle\] \[\left\vert \psi_2\right\rangle = \frac{4}{5}\left\vert +\right\rangle- i\frac{3}{5} \left\vert -\right\rangle\] \[\left\vert \psi_3\right\rangle = -\frac{4}{5}\left\vert +\right\rangle+ i\frac{3}{5} \left\vert -\right\rangle\]
      1. For each of the \(\left|{\psi_i}\right\rangle \) above, calculate the probabilities of spin component measurements along the \(x\), \(y\), and \(z\)-axes.
      2. Look For a Pattern (and Generalize): Use your results from \((a)\) to comment on the importance of the overall phase and of the relative phases of the quantum state vector.
    • group Directional Derivatives

      group Small Group Activity

      30 min.

      Directional Derivatives
      Vector Calculus I 2022

      Directional derivatives

      Gradient Sequence

      This small group activity using surfaces relates the geometric definition of directional derivatives to the components of the gradient vector. Students work in small groups to measure a directional derivative directly, then compare its components with measured partial derivatives in rectangular coordinates. The whole class wrap-up discussion emphasizes the relationship between the geometric gradient vector and directional derivatives.
    • group The Hillside

      group Small Group Activity

      30 min.

      The Hillside
      Vector Calculus I 2022

      Gradient Sequence

      Students work in groups to measure the steepest slope and direction on a plastic surface, and to compare their result with the gradient vector, obtained by measuring its components (the slopes in the coordinate directions).
    • assignment Symmetry Arguments for Gauss's Law

      assignment Homework

      Symmetry Arguments for Gauss's Law
      Static Fields 2022 (4 years)

      Instructions for 2022: You will need to complete this assignment in a 15 minute appointment on Zoom or in person with one of the members of the teaching team between 1/21 and 10 pm on 1/26. Here is a link to a sign-up page.

      You are required to watch a sample video for how to make symmetry arguments here. As demonstrated in the video you should bring with you to the meeting a cylinder, an observer, and a vector.

      Use good symmetry arguments to find the possible direction for the electric field due to a charged wire. Also, use good symmetry arguments to find the possible functional dependence of the electric field due to a charged wire. Rather than writing this up to turn in, you should find a member of the teaching team and make the arguments to them verbally.

  • Static Fields 2022 (5 years)

    You are on a hike. The altitude nearby is described by the function \(f(x, y)= k x^{2}y\), where \(k=20 \mathrm{\frac{m}{km^3}}\) is a constant, \(x\) and \(y\) are east and north coordinates, respectively, with units of kilometers. You're standing at the spot \((3~\mathrm{km},2~\mathrm{km})\) and there is a cottage located at \((1~\mathrm{km}, 2~\mathrm{km})\). You drop your water bottle and the water spills out.

    1. Plot the function \(f(x, y)\) and also its level curves in your favorite plotting software. Include images of these graphs. Special note: If you use a computer program written by someone else, you must reference that appropriately.
    2. In which direction in space does the water flow?
    3. At the spot you're standing, what is the slope of the ground in the direction of the cottage?
    4. Does your result to part (c) make sense from the graph?

  • Media & Figures
    • figures/x2y3d.png
    • figures/x2yC.png
    • figures/x2yC2.png