Students explore path integrals using a vector field map and thinking about integration as chop-multiply-add.
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 Small Group Activity
30 min.
group Small Group Activity
30 min.
group Small Group Activity
30 min.
group Small Group Activity
30 min.
vector calculus coordinate systems curvilinear coordinates
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 Small White Board Question
10 min.
vector differential rectangular coordinates math
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 Small Group Activity
30 min.
assignment Homework
face Lecture
5 min.
group Small Group Activity
30 min.
group Small Group Activity
30 min.