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Section 2.3 Line Integral of a Gradient Vector Field

We've so far seen line integrals and vector fields. Now we put them together and investigate the line integral of a vector field along a curve. In physics and engineering, when the fields are force fields like gravity, this will capture the concept of Work.
Learning Objectives:
  1. Geometrically describe the line integral of a vector field
  2. Connect the physics concept of work to a line integral
  3. State a definition for the line integral of a vector field
  4. State a formula to compute the line integral of a vector field given a parameterization of the curve

Post-Video Activities Post-Video Activities

1.

One application of line integrals of vector fields I mentioned already: work. This is when the field represents a force. What if the field represented a velocity field, like a swirling body of water with each vector showing the velocity of water at that point. What type of thing would the same line integral formula represent physically in that case? Note: you might not have a nice name for it like we did for work just yet, we'll give a name later:)

2.

I LOVE this animation of fluid flows 1  Note that while in this course we primarily focus on static fields (i.e they don't change in time) this animation let's you see what happens with dynamic fields that get perturbed by your mouse input.
paveldogreat.github.io/WebGL-Fluid-Simulation/