## Student handout: Vector Integrals (Contour Map)

Students explore path integrals using a vector field map and thinking about integration as chop-multiply-add.
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.

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.

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 $\vec{F})$.

• 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 Number of Paths

group Small Group Activity

30 min.

##### 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.
• 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 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 Covariation in Thermal Systems

group Small Group Activity

30 min.

##### Covariation in Thermal Systems

Students consider how changing the volume of a system changes the internal energy of the system. Students use plastic graph models to explore these functions.
• assignment The Path

assignment Homework

##### The Path

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?
• 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 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 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.

Keywords
E&M Path integrals
Learning Outcomes