Mathematics is used in almost every aspects of life. Natural phenomenon and processes are described by mathematics. Sciences and social sciences apply mathematics to help describe processes and quantify values.
Application of mathematics in everyday life ranges from simple arithmetic calculations to the most advanced calculations like calculus and trigonometry.
In this post I would like to share this information that most advanced function in mathematics and physics are derived from simple fundamental principles of mathematical. Yes they are derived from simple everyday maths rules so remember that.
For example:
Derivative calculus applies the principal of subtraction and division. (X1 - X2)/(Y1-Y2)
Pythagoras Theorem applies the summation of simple algebra: A squared = B squared + C squared
Pressure calculations apply division: Force/Area
Force calculations apply multiplication: Mass x Acceleration
etc. to name a few.
The above are few function in mathematics and physics in their simplest form. They are the fundamental equations that many many advanced equations can be derived from (the same applies to other subjects).
Advanced and complex maths functions a formed singly from a simple mathematics concept or derived as a compound of two or more principles working in sync.
Pay attention to the example below to illustrate this concept (read with an open mind).
The equation used to determine factor of safety of a planar failure in geotechnical engineering in a more general form.
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| Figure 1. Plane Slope |
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| Figure 2. Factor of Safety Equation |
The equations above (figure 2) come with symbols and notations that makes it seem interestingly fancy and somewhat complicated to understand. However following is the analysis of the derivation that should help you understand that even a fancy looking formula (like the one above or others that you encounter) with a lot of symbols and functions were derived from basic mathematics and subject respective principles:
Simple Analysis Steps:
1. Define the function properly:
Define the function in its simple terms and what it means. In this case Factor of safely
For example:
FOS (aka Factor of Safety) is basically a ratio (application of division).
Its (FOS) defined as the Resisting force (force resisting a plane to slide) divided by Driving Force (force pushing the plane to slide)
2. Identify different components that make up the definition above (1).
For Instance: Identifying Driving Force components (eg: Depth of Tension Crack, Weight of unstable block, Water uplifting force, Driving water force) and Resisting forces (friction angle, slope angel, weight of resisting block etc)
3. Identify individual Operation and Order of Operation
For instance: Division, Summation, Trigonometric identities etc.
4. Execute function
Calculating Factor of Safety (FOS) by applying principles of addition, subtraction, multiplication and division in line with the definition (1).
The derivation is as follows:
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| Figure 3: Derivation of FOS of plane failure |
Note: You are not expected to derive or know this formulas but to visualize how simple maths rules are applied here.
Insight
The derivation comes from breaking down the illustration above into two broad groups of forces (resisting and driving force). All resisting forces are summed on the numerator and all driving forces are summed as divisor. Finally the division of the two gives the FOS. Other symbols and notations are trigonometric identities and pro-numerals used in breaking down the force components with respect to the sliding plane angle.
Now you can realize that this fancy looking equation was derived from simple division, summation and trigonometry.
Summary
In summary all complex mathematical functions in a nutshell employ simple maths and science rules that we tend to ignore. The best practice is to firstly break down the fancy equation to simple everyday maths that makes sense to you before proceeding to analysing the components of the equation. Of course it requires your sound logical reasoning skills, analysis and background understanding and know-how of the respective subject you're working on.
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