## 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

• Red quadrupole surface
• 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 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 Integrals (Contour Map)

group Small Group Activity

30 min.

##### Vector Integrals (Contour Map)

• group The Hill

group Small Group Activity

30 min.

##### The Hill
Vector Calculus II 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 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 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.
• assignment_ind Vector Differential--Rectangular

assignment_ind Small White Board Question

10 min.

##### Vector Differential--Rectangular
Static Fields 2022 (7 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 Vector Differential--Curvilinear

group Small Group Activity

30 min.

##### Vector Differential--Curvilinear
Vector Calculus II 2022 (8 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.

• group The Hillside

group Small Group Activity

30 min.

##### The Hillside
Vector Calculus I 2022

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).
• 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.
• group Static Fields Equation Sheet

group Small Group Activity

5 min.

##### Static Fields Equation Sheet
Static Fields 2022 (3 years)

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