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Can somebody please explain how partial derivatives are used and where they can be...

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Wilson2014 | eNotes Employee

Posted May 24, 2013 at 11:59 PM via web

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Can somebody please explain how partial derivatives are used and where they can be applied in a real life situation/problem? 

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llltkl | College Teacher | (Level 3) Valedictorian

Posted May 25, 2013 at 1:00 AM (Answer #1)

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 In differential calculus, the derivative of a function of several variables with respect to one of them and with the remaining variables treated as constants is called partial derivative. The partial derivative of a function f with respect to the variable x is variously denoted by

f'x,  f,x,   ∂xf,   or ∂f/∂x

The partial-derivative symbol is ∂. Partial derivatives are used in vector calculus and differential geometry.

Example :       Let f(x,y)=y^3x^2. Calculate ∂f/x(x,y).

Solution: To calculate ∂f/∂x(x,y), we simply view y as being a fixed number and calculate the ordinary derivative with respect to x. The first time you do this, it might be easiest to set y=b, where b is a constant, to remind you that you should treat y as though it were number rather than a variable. Then, the partial derivative ∂f/x(x,y) is the same as the ordinary derivative of the function g(x)=b^3x^2. Using the rules for ordinary differentiation, we know that

`(delg)/(delx)(x)=2b^3x.`

Now, we remember that b=y and substitute y back in to conclude that

`(delf)/(delx)(x,y)=2y^3x.`

Partial differential equations abound in all branches of science and engineering and many areas of business. The number of applications is endless. In real-life engineering, you have software programs using numerical techniques like finite differences to solve the partial derivative problems. Equations involving partial derivatives are known as partial differential equations (PDEs) and most equations of physics are PDEs:

(1) Maxwell's equations of electromagnetism
(2) Einstein's general relativity equation for the curvature of space-time given mass-energy-momentum.
(3) The equation for heat conduction (Fourier)
(4) The equation for the gravitational potential of a blob of mass (Newton-Laplace)
(5) The equations of motion of a fluid (gas or liquid) (Euler-Navier-Stokes)
(6) The Schrodinger equation of quantum mechanics
(7) The Dirac equation of quantum mechanics

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