## Activity: Number of Paths

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.
What students learn
• Various differences between conservative and non-conservative fields
• Line integrals on conservative fields are path independent.
• Conservative fields are gradients of a scalar potential that can be represented as a surface; non-conservative fields cannot.
• Media

Number of Paths: For each of the fields, mark two different points $A$ and $B$. Can you identify paths where the vector line integral has the indicated value (greater than, equal to, or less than zero)? In the table, enter the number of paths you can find for each condition (max 2 paths). Discussion: Breaking into pieces Some students don't realize you can do piecewise paths (a curve connected to a straight line). Talk to these students right away.

Discussion: Choosing easier paths Discuss what paths are the most convenient for estimating the sign of the path integral and why.

Optional Further Discussion: A third field $\vec{F}_{3}$ could also be given to students. Its curl is zero everywhere except the center. It corresponds to a magnetic field for an infinite current carrying wire.

Match a Surface: One of the vector fields corresponds to (part of) your surface. Where does it match, and how do you know?

Student Reasoning: The gradient is perpendicular to the level curves and points in the direction of increasing function. The line integral corresponds to a change in height on the surface.

Discussion: Gradient and Field Some students need to be told that the vector field is the gradient. The gradient will point in the direction of increase. Note that if the field is the electric field and the surface is the potential, there is a relative minus sign so that the field points in the direction of decreasing potential. This ideas is brought out much more strongly in the Work and Electric Field activity.

Extend to a New Surface: Could the other vector fields correspond to a surface? Explain why or why not.

They could not, at least not in the typical way. This question can generate some interesting discussions about what information is encoded in scalar field.

SUMMARY PAGE
What Students Learn:
• Some differences between conservative and non-conservative fields
• Line integrals on conservative fields are path independent.
• Conservative fields are gradients of a scalar potential that can be represented as a surface; non-conservative fields cannot.

Time Estimate: 30 minutes

Equipment

• Vector field handout in dry-erase sleeve for each group
• Dry-erase markers & erasers
• Whiteboard for each group
• Student handout for each student

Introduction

• Students should have some practice with vector line integrals. We suggest Vector Integrals as a warmup activity.

Whole Class Discussion / Wrap Up:

• Ask students to explain their reasoning and report their findings. Look for consensus/disagreement in the reasoning.

• group Vector Integrals (Contour Map)

group Small Group Activity

30 min.

##### Vector Integrals (Contour Map)

Students explore path integrals using a vector field map and thinking about integration as chop-multiply-add.
• group Work By An Electric Field (Contour Map)

group Small Group Activity

30 min.

##### Work By An Electric Field (Contour Map)

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 Vector Differential--Curvilinear

group Small Group Activity

30 min.

##### Vector Differential--Curvilinear
Vector Calculus II 23 (11 years)

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.

• assignment_ind Vector Differential--Rectangular

assignment_ind Small White Board Question

10 min.

##### Vector Differential--Rectangular
Vector Calculus II 23 (10 years)

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..

• group Heat and Temperature of Water Vapor

group Small Group Activity

30 min.

##### Heat and Temperature of Water Vapor

In this introduction to heat capacity, students determine a derivative that indicates how much the internal energy changes as the temperature changes when volume is held constant.
• group Quantifying Change

group Small Group Activity

30 min.

##### Quantifying Change

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.
• group Changes in Internal Energy (Remote)

group Small Group Activity

30 min.

##### Changes in Internal Energy (Remote)

Warm-Up

Students consider the change in internal energy during three different processes involving a container of water vapor on a stove. Using the 1st Law of Thermodynamics, students reason about how the internal energy would change and then compare this prediction with data from NIST presented as a contour plot.
• face Unit Learning Outcomes: Classical Mechanics Orbits

face Lecture

5 min.

##### Unit Learning Outcomes: Classical Mechanics Orbits
Central Forces 2023 This handout lists Motivating Questions, Key Activities/Problems, Unit Learning Outcomes, and an Equation Sheet for a Unit on Classical Mechanics Orbits. It can be used both to introduce the unit and, even better, for review.
• group Gravitational Potential Energy

group Small Group Activity

60 min.

##### Gravitational Potential Energy

Students examine a plastic “surface” graph of the gravitational potential energy of an Earth-satellite system to explore the properties of gravitational potential energy for a spherically symmetric system.
• group Electric Field of Two Charged Plates

group Small Group Activity

30 min.

##### Electric Field of Two Charged Plates
• Students need to understand that the surface represents the electric potential in the center of a parallel plate capacitor. Try doing the activity Electric Potential of Two Charged Plates before this activity.
• Students should know that
1. objects with like charge repel and opposite charge attract,
2. object tend to move toward lower energy configurations
3. The potential energy of a charged particle is related to its charge: $U=qV$
4. The force on a charged particle is related to its charge: $\vec{F}=q\vec{E}$

Author Information
Liz Gire, Aaron Wangberg, Robyn Wangberg, and the Surfaces team
Keywords
E&M Conservative Fields Surfaces
Learning Outcomes