M408M Learning Module Pages

Main page

Chapter 10: Parametric Equations and Polar Coordinates

Chapter 12: Vectors and the Geometry of Space


Chapter 13: Vector Functions

Chapter 14: Partial Derivatives

Learning module LM 14.1: Functions of 2 or 3 variables:

Learning module LM 14.3: Partial derivatives:

Learning module LM 14.4: Tangent planes and linear approximations:

Learning module LM 14.5: Differentiability and the chain rule:

Learning module LM 14.6: Gradients and directional derivatives:

       Gradients
      Gradients and hill climbing
      Wind and weather
      Directional derivatives
      Worked problems

Learning module LM 14.7: Local maxima and minima:

Learning module LM 14.8: Absolute maxima and Lagrange multipliers:

Chapter 15: Multiple Integrals

Wind and weather

Wind and Weather Atmospheric Pressure, Wind, and Gradients: Let $z = P(x,\, y)$ be the pressure at the point $(x,\,y)$ in some region $D$ in the $xy$-plane (flat earth!). Then $\nabla P$ is often referred to as the Pressure Gradient, not surprisingly!

The level curves are curves of constant pressure, called isobars, and a standard weather map for $D$ consists of a set of such isobars . As a visual aid this contour map colors are often used to indicate pressure at a given isobar - the darker the color, the lower the pressure. High pressure areas on weather maps are generally associated with clear skies, while low pressure areas are generally associated with cloudy or overcast skies. Wind is generated by the differences in pressure, as the air feels a force proportional to $- \nabla P$. The bigger the pressure gradient, the stronger the wind.

However, once it gets started the wind does not continue to move in the direction of minus the pressure gradient! Its trajectory gets bent 90 degrees sideways (right in the Northern Hemisphere, left in the Southern Hemisphere) by the Coriolis force, a consequence of the Earth's rotation. Since gradients are perpendicular to level curves, this means that wind generally moves along isobars. (In the Northern Hemisphere, the wind circulates clockwise around high pressure systems and counter-clockwise around low pressure systems.) Since storms are associated with low pressure, most (Northern Hemisphere) storms rotate counter-clockwise around their centers. Remember that the next time you read about a hurricane! So where will the wind be blowing and how strong will it be blowing in the weather map to the right? Move the slider to rotate the vector field. How many radians do you have to rotate from the gradient to get a realistic wind pattern?