Expert: Scotto - 9/23/2011

Question
QUESTION: I want to ask when is partial derivative applied?Is it when there is a function of more than one variables ?
Also for what purpose it is applied ie. for finding slope of tangent  
or any other application

ANSWER: The Master's paper I wrote in college dealt with partial derivatives.  It was on the effect of shock wave and rarefaction fans.  A shockwave occurs when to pieces of air are suppose to occupy the same place at the same time.  They can't do that, so they generate a shock wave.  A rarefaction fan is generated when the space is suppose to be a vaccuum.  Gas can't do that either, so when this occurs, there is turbulence that is generated when all of the air rushes in to fill it.  The shape of airplane wings does this and that is what lets them fly.

When dealing with derivatives, there is the partial derivative with respect to time, and three partial derivatives with respect to up-down, forward-backward, and left-right.

In physics, most problems deal with 2 dimensional space.  One is for how far away and the other is for how high.  The partial derivatives can be taken in terms of either one.  In most cases, horizontal force is man-made, whereas vertical force is for gravity.  The derivative of acceleration is speed, and the derivative of speed is distance.  The partial derivatives occur when looking only at one direction.

When dealing with forces, partial derivatives can be taken in any direction to see the affects in that direction.


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QUESTION: I must say A VERY COMPLEX SOLUTION!!!.I could not understand it plzzzz explain clearly

Answer
Use of Differential Equations
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When objects move thru the air, the function describing they're position is f(x,y,z,t).
The variables x, y, and z are for the height, width, and depth, and t is for time.
Basically, a partial derivative with respect to x is a derivative of the function treating all other variables as constants.

Master's Paper
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In my Master's paper, I only dealt clearly with one dimension and time, but referred to three dimensions.  The methods I used were the straightforward (which didn't do so well, for it generated a ripple affect.  The others methods I researched were by Lax-Wendroff, Gudonov, Enquist-Osher, but that was finished back in 1986.

As functions move thru time, they have a force which motivates them.  A rarefaction fan examplee was a fairly simple example of where the force for negative x was non-existant and the force on the positive x was +1.  A force of this nature generates a vaccuum as time progresses since the particles on the left side of 0 aren't moving and the particles on the right size or veering away.  Mathematically, this would create a vaccum at 0 that was widening to (0,t) in in the x-space.  This was because the particles on the left side were moving to the right at a steady pace of 1.

Another problem I looked at was one where the particles on the negative x axis had a motion of 1 and the particles on the right were still.  When this occurs in real-life (in 3 dimensions), a shock wave is generated.

The mehtods I mentioned tried to detect shock waves and rarefaction fanes using numercial analysis to determine wheth to generate the motion at the next x coordinate from the right or left, though none of them could do it exactly.

Distance, Velocity, Acceleration
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As far as derivatives go, there were partical derivatives involved with respect to distance and time.  The derivative with respect to time gave a reference to where a paritcal was located and the partial derivative with respect to position gave how fast data was moving thru that point.

5th Order Differential Equations
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To undestand the application of derivates, much more in depth work is needed.  Back in college, I dealt with 2nd order differential equations.  To apply these, note that speed is the derivative of distance and acceleration is the derivative of speed.  Since then, I have solved up to 5th order differential equations.  To solve them involves solving the polynomial generated.
For example, if y""' is the 5th derivative of y with respect to x, then to solve
y""' + 3y"" - 3y"' - 9y" + 2y' + 6, first look at what is called the characteristic equation.
That is, x^5 + 3x^4 - 3x^3 - 9x^2 + 2x + 6.  This factors out to be (x+1)(x+2)(x+3)(x-1)(x-2).
The solutions to that equation are -1, -2, -3, 1, and 2.  This means the equation is
Ae^(-x) + Be^(-2x) + Ce^(-3x) + De^(x) + Ee^(2x) for some constants A, B, C, D, and E.

My Current Work
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Now as far as why I'm not working at a job, I was in a very serious automobile accident.  I was in a Datsun 510 (slightly larger than a VW) and turned left off a highway only to have a one ton pickup come shooting out of nowhere and hit me.  According to the doctors, the left side of my brain was killed, and that is suppose to control mathematics and the right side of my body.  I was only given a 2% chance of living, with no chance of walking.  There was 0% chance of me ever being able to ride a bike.

However, after trusting in the Lord and Memorizing chapters out of the Bible, I can still remember work that was done before the accident quite clearly, though it is hard to learn new.
I did work at Boeing where everyone had a Master's Degree, but was laid off in the 90's when they were downsizing.  I was hired without even looking by Safeway in Oregon back when they counted bottles by hand, but had to quit that after my feet broke out with Planter's Fasciitis.
That make my seet unable to stand in one spot or support me for more than a couple of hours.

Recent Experience
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Since then, I have been a stay at home dad to over 50 foster children (no more than 4 at once), playing the piano on a voluntary basis for hundreds of songs off the top of my head plus a few I have written.  I have also taking to reading the Bible and praying for many people and events.  The church I am attending has had people healed by prayer every week for several years.  Also, God has rewarded us for obeying the Bible by freeing us from several things that could have gone wrong.  For one example, my wife drives a car we purchased many years ago that requires little maintenance.  As another, the man upgrading our kitchen refers to a as his miracle clients.  As a few examples, there was a section of our house with dry-rot that stopped about the time we bought the house and whatever he needs to buy happens to be on sale.