Wind – Chill Index “A Calculus Approach” By Felix Garcia

Rationale In regions with severe winter weather, the wind – chill index is often used to describe the apparent severity of the cold. This index W is a subjective temperature that depends on the actual temperature T and the wind speed v. So W is a function of T and v and we can write

The purpose of this presentation is to show that we can use multivariable calculus to calculate rates and approximations using only “raw data”, i.e. No empirical function is given. No closed formula from a model or theory is known. No regression analysis is used to obtain a formula that allows us to do calculus. Instead, we use values of W compiled by the NOAA (National Oceanographic and Atmospheric Administration) and the Meteorological Service of Canada. The following table is an excerpt from that data.

V T Wind speed (km/h) Actual Temperature (˚C) The values in the table is the perceived temperature W when the actual temperature is T and the wind speed is v. For example, if T = -15˚C and v = 50 km/h the subjective temperature is -29˚C ( the intersection of the row that corresponds to -15˚C and the column that corresponds to 50 km/h.

Actual Temperature (˚C) Subjective Temperature (˚C) Chart of subjective (perceived) temperature

An important question in multivariable calculus is: Which is the rate of the change of the dependent variable W with respect to change in one of the independent variables T or v ? This rate is called the partial derivative of W with respect to T or v. In order to make an estimate of those partial derivatives we consider two rates centered at the given point and we take the average of both. For example …

To estimate we calculate and we take the average of both of them. In the vertical direction, h = 5 therefore

Similarly, in the horizontal direction, h=10 so are the two rates that we must average to estimate

A second question in multivariable calculus is: How we can use the linearization of the function determined by the table to make approximations to values not shown in the table? The linearization L(T,v) is by definition the tangent plane to the function at the given point, i.e. Finally,

The purpose of this linearization is to approximate Assume that we want to estimate W(-17,42) using the linearization. We have Therefore,

A note on Discretization Discretization concerns the process of transferring continuous models and equations into discrete counterparts. This process is usually carried out as a first step toward making them suitable for numerical evaluation and implementation on digital computers. The image at left shows a solution to a discretized partial differential equation, obtained with the finite element method.

Summary This is an example of the so-called discretization of the continuous that started with the introduction of computers into everyday life. Without the use of a continuous model and the traditional machinery of calculus we can derive good estimates that a few decades ago were unheard of.