Divergence and Curl: Apply It

  • Determine divergence from the formula for a given vector field
  • Determine curl from the formula for a given vector field
  • Use divergence and curl to identify conservative fields

Analyzing Airflow in Your Dorm Room

It’s the first hot week of the semester, and your small dorm room feels stuffy despite having two fans running. Your roommate, who is studying mechanical engineering, offers to help optimize your airflow setup. They place sensors at various points in the room and model the combined air velocity as vector fields based on different fan positions. The velocity vectors describe how fast and in what direction air moves at each point in the room. By analyzing these vector fields, you can determine where air is stagnating, where it’s flowing effectively, and whether the fans create useful circulation patterns. Understanding these airflow characteristics will help you reposition the fans for maximum cooling comfort during the hot days ahead.

Your roommate’s first sensor reading models the airflow velocity in a horizontal cross-section of the room as the vector field [latex]\mathbf{v}(x, y) = \langle 3x + y,  2x - 3y \rangle[/latex], where [latex]x[/latex] and [latex]y[/latex] are measured in meters from the center of the room, and velocity is in meters per second.

Your roommate repositions the oscillating fan and the new airflow in a vertical cross-section is modeled by the three-dimensional vector field [latex]\mathbf{F}(x, y, z) = \langle yz, xz,  xy \rangle[/latex], where [latex]z[/latex] represents height above the floor.

Your roommate tries a different fan configuration. The resulting airflow is modeled by [latex]\mathbf{G}(x, y, z) = \langle 2xz,  y^2, x^2 + z \rangle[/latex].

You want to know if certain airflow patterns could represent a conservative vector field, which would mean the air circulation is “path-independent” and easier to predict. Your roommate tests two different fan setups:

Your roommate proposes one final airflow model: [latex]\mathbf{H}(x, y, z) = \langle e^x \sin y,  e^x \cos y,  z^2 \rangle[/latex].