Activity: Vector Integrals (Contour Map)

What students learn
  • Recall the relationship between the sign of the dot product and the orientation of the vectors.
  • Use graphical methods to estimate the value of a vector line integral.
  • Lay the groundwork for thinking about conservative and non-conservative vector fields.
  • Media
    • activity_media/NewVectorPlot.pdf
    • vector2.png
    • NewVectorField.PNG

The Sign of Work: For the force vector shown above, draw three paths such that the work done by \(\vec{F}\) is positive, zero, and negative for small displacements along each path.

Goal: This gets at the dot product relationship students will need to perform the vector path integral. It does not get at the ideas that you have to chop the whole path into small pieces to do the integral (see next question).

Follow-Up: “Is there only one correct path for each?” Students usually draw the parallel/antiparallel/perpendicular paths and do not consider other options.


Estimate Vector Line Integral: For each segment of the path on the vector field \(\vec{F}\) shown below, estimate the value of the integral:

\[ \int_{\textrm{path}} \vec{F} \cdot d\vec{r} \]

where the side of each square is 1 cm and the length of the longest arrow is 10 units (appropriate for the field \(\vec{F})\).

Discussion: Units Students might ask about the units of \(\vec{F}\). This is a great place to talk about how for an integral, the units of the result are the units of the integrand times the units of the infinitesimal (students often ignore the units of the infinitesimal). Also, for E&M, students are going to do a similar integral with the electric field - which does not have units of force - so being mindful about the units of the field is important.

Answer: We particularly recommend students use the Chop, Multiply, Add strategy, but there are other reasonable methods here.

For the leftward path, students need to be able to chop the path into segments (we recommend 4 segments), find the value of the dot product for that path, and then add up those four values to get the line integral.

For the upward path, the path is always perpendicular to the vector field, so the line integral is zero.

For the diagonal path (the hardest path), students need to chop up the path and consider the different magnitude at each point, but they also need to account for the nontrivial angle between the path and the vector field. This can be done in several ways---students will often try to find an explicit form for \(\vec{dr}\), but they can also use a geometric argument to use only the vertical component of the path.


SUMMARY PAGE
What Students Learn:
  • Recall the relationship between the sign of the dot product and the orientation of the vectors.
  • Use graphical methods to estimate the value of a vector line integral.
  • Lay the groundwork for thinking about conservative and non-conservative vector fields.

Time Estimate: 15 minutes

Equipment

  • Dry-erase markers & erasers
  • Whiteboard for each group
  • Student handout for each student

Introduction

  • This activity is a warmup activity and students do not need an introduction.

Whole Class Discussion / Wrap Up:

  • This activity is a warmup to two other activities: Number of Paths and Work by an Electric Field. We highly recommend doing this activity before either of these activities.

  • Because there are multiple possible answers for the first part of the activity, and multiple possible strategies for evaluating the integral in the second part, this activity is particularly well suited to a whole-class discussion in which different groups of students are asked to share their answers and discuss the advantages and disadvantages of each.

  • group Work By An Electric Field (Contour Map)

    group Small Group Activity

    30 min.

    Work By An Electric Field (Contour Map)

    E&M Path integrals

    Students will estimate the work done by a given electric field. They will connect the work done to the height of a plastic surface graph of the electric potential.
  • group Number of Paths

    group Small Group Activity

    30 min.

    Number of Paths

    E&M Conservative Fields Surfaces

    Student discuss how many paths can be found on a map of the vector fields \(\vec{F}\) for which the integral \(\int \vec{F}\cdot d\vec{r}\) is positive, negative, or zero. \(\vec{F}\) is conservative. They do a similar activity for the vector field \(\vec{G}\) which is not conservative.
  • assignment The Path

    assignment Homework

    The Path

    Gradient Sequence

    Vector Calculus I 2022 You are climbing a hill along the steepest path, whose slope at your current location is \(1\over5\). There is another path branching off at an angle of \(30^\circ\) (\(\pi\over6\)). How steep is it?
  • group Vector Differential--Curvilinear

    group Small Group Activity

    30 min.

    Vector Differential--Curvilinear
    Vector Calculus II 2022 (8 years)

    vector calculus coordinate systems curvilinear coordinates

    Integration Sequence

    In this small group activity, students are given a picture as a guide. They then write down an algebraic expression for the vector differential in different coordinate systems (cartesian, cylindrical, spherical).

    Use Vector Differential--Rectangular as an introduction. This activity can be done simultaneously with Pineapples and Pumpkins where students or the instructor cut volume elements out of pineapples and/or pumpkins to show the geometry.

  • group Quantifying Change

    group Small Group Activity

    30 min.

    Quantifying Change

    Thermo Derivatives

    In this activity, students will explore how to calculate a derivative from measured data. Students should have prior exposure to differential calculus. At the start of the activity, orient the students to the contour plot - it's busy.
  • assignment_ind Vector Differential--Rectangular

    assignment_ind Small White Board Question

    10 min.

    Vector Differential--Rectangular
    Static Fields 2022 (7 years)

    vector differential rectangular coordinates math

    Integration Sequence

    In this introductory lecture/SWBQ, students are given a picture as a guide. They then write down an algebraic expression for the vector differential in rectangular coordinates for coordinate equals constant paths.

    This activity can be done as a mini-lecture/SWBQ as an introduction to Vector Differential--Curvilinear where students find the vector differential in cylindrical and spherical coordinates..

  • assignment The puddle

    assignment Homework

    The puddle
    differentials Static Fields 2022 (4 years) The depth of a puddle in millimeters is given by \[h=\frac{1}{10} \bigl(1+\sin(\pi xy)\bigr)\] Your path through the puddle is given by \[x=3t \qquad y=4t\] and your current position is \(x=3\), \(y=4\), with \(x\) and \(y\) also in millimeters, and \(t\) in seconds.
    1. At your current position, how fast is the depth of water through which you are walking changing per unit time?
    2. At your current position, how fast is the depth of water through which you are walking changing per unit distance?
    3. FOOD FOR THOUGHT (optional)
      There is a walkway over the puddle at \(x=10\). At your current position, how fast is the depth of water through which you are walking changing per unit distance towards the walkway.
  • 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.
  • face Warm-Up Powerpoint

    face Lecture

    10 min.

    Warm-Up Powerpoint

    Warm-Up

    The attached powerpoint articulates the possible paths through the curriculum for new graduate students at OSU.
  • assignment Paramagnet (multiple solutions)

    assignment Homework

    Paramagnet (multiple solutions)
    Energy and Entropy 2021 (2 years) We have the following equations of state for the total magnetization \(M\), and the entropy \(S\) of a paramagnetic system: \begin{align} M&=N\mu\, \frac{e^{\frac{\mu B}{k_B T}} - e^{-\frac{\mu B}{k_B T}}} {e^{\frac{\mu B}{k_B T}} + e^{-\frac{\mu B}{k_B T}}}\\ S&=Nk_B\left\{\ln 2 + \ln \left(e^{\frac{\mu B}{k_B T}}+e^{-\frac{\mu B}{k_B T}}\right) +\frac{\mu B}{k_B T} \frac{e^{\frac{\mu B}{k_B T}} - e^{-\frac{\mu B}{k_B T}}} {e^{\frac{\mu B}{k_B T}} + e^{-\frac{\mu B}{k_B T}}} \right\} \end{align}
    1. List variables in their proper positions in the middle columns of the charts below.

    2. Solve for the magnetic susceptibility, which is defined as: \[\chi_B=\left(\frac{\partial M}{\partial B}\right)_T \]

    3. Using both the differentials (zapping with d) and chain rule diagram methods, find a chain rule for:

      \[\left(\frac{\partial M}{\partial B}\right)_S \]

    4. Evaluate your chain rule. Sense-making: Why does this come out to zero?


Keywords
E&M Path integrals
Learning Outcomes