Understanding Calculus

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  Table of Contents

  1. Why Study
  2. Numbers
  3. Functions
  4. The Derivative
  5. Differentiation
  6. Applications
  7. Free Falling
  8. Understanding
  9. Derivative
  10. Integration
  11. Understanding
  12. Differentials

  Inverse Functions
  Applications of
  Sine and Cosine
  Sine Function
  Sine Function -
  Differentiation and
  Oscillatory Motion
  Mean Value
  Taylor Series
  More Taylor Series


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Chapter 3 - The Mathematical function and its Graph

Section 3.3 - Dimensions

The mathematical function is a relationship that defines a situation in terms of a set of interacting conditions. Neither the situation nor the condition can exist independent of each other. The function states how a set of fixed conditions come together to define a situation. In mathematics, conditions are denoted by the term, dimension. Before one can understand the concept of the function, we need to explain what a dimension is.

In science there are four fundamental dimensions used to describe any physical situation in the natural world. They are mass, length, temperature and time. While mass, length and temperature refer to static situations, time is a dynamic dimension used to describe changes that define an action.

Think of time as a passage of events. For example a summer spent doing nothing but eating, sleeping and watching TV would seem to have gone by in a few days. On the other hand a summer spent traveling around the world would seem to have lasted forever. If we assume each vacation to have been 80 days, ones memory of the uneventful vacation consists only of a few events that could have been done in a day whereas the vacation around the world consisted of a plethora of events, such that relative to the boring summer, they took longer to finish.

For time to pass something must change with respect to itself. The units of time, the second, is nothing more than a reference for any change. If one second refers to a pendulumís swing through one arc, then any changing dimension can be measured relative to the standard of one second. For example a moving carís distance (length) from a reference point is constantly changing. The changing distance can only be measured with respect to time.

It is important to understand that time is the dimension for actions, while length, mass and temperature refer to static conditions. If length, mass or temperature were changing with respect to itself then the change would have to be analyzed with respect to time. To summarize, for time to pass some action must occur and for an action to occur some dimension must be changing with respect to itself.

Length, mass, temperature, and time refer to the simplest absolute dimensions the physical world can be reduced to. Other dimensions derived from these fundamental dimensions include, force (dependent on mass), stress (dependent on force), elasticity (dependent on force and stress), velocity (dependent on length and time), kinetic energy (dependent on mass and velocity), etc.

Many derived dimensions are dependent on other derived dimensions such as kinetic energy is dependent on velocity which is dependent on time. But what about people, dollars, etc. Clearly these are also measurable conditions; however, they can not be easily expressed in terms of the fundamental dimensions of mass, length, time, and temperature. Since they are unique dimensions we need to come up with a consistent definition for a dimension that applies to all fundamental, derived, and unique dimensions.

A dimension is simply a quantifiable condition that describes a situation. In science we always study situations where external factors are removed from the analysis. It is the objects that remain along with their properties that become our focus of study. A dimension is a measurement of a property that is relevant to the situation being analyzed. By themselves, dimensions are meaningless. They must refer to certain conditions specific to the situation being studied. One cannot refer to just mass or length. One has to specify mass and length of what part of the system? The units of the various dimensions, meter, seconds, Celsius, kg, serves as standards to measure the dimension relative to.

Understanding which dimensions to include in your scientific analysis all depends on the type of situation being studied. The more familiar you are with the fundamental and derived dimensions, the easier it will be for you to understand which dimensions are significant and how they define the situation.

Next section -> Section 3.4 - The Mathematical Function


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