Wind Chill Index A Calculus Approach

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Wind Chill IndexA Calculus Approach

• By
• Felix Garcia

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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

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• 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.

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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
-15C and v 50 km/h the subjective temperature
is -29C ( the intersection of the row that
corresponds to -15C and the column that
corresponds to 50 km/h.
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Chart of subjective (perceived) temperature
Subjective Temperature (C)
Actual Temperature (C)
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• 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

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

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• Similarly, in the horizontal direction, h10 so
  are the
  two
• rates that we must average to estimate

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• 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,

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

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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.
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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.