10.Chemistry & Mathematics VI-Calculus(3)-Differentiation(3).

Partial Differentiation.

In this post I intend to discuss the concept of PARTIAL DIFFERENTIATION.We have already studied that for a function with one variable ‘y’ , y=f(x), the derivative

f'(x) = dy/dx  ⇒ represents the rate of change of ‘y’ as ‘x’ changes.

But what if there are more than one variables?

eg. In thermodynamics, Pressure(P),Volume(v),Temperature(T),Internal Energy (E),Entropy(S) → all these quantity define a thermodynamic system.Most of the times, more than one of these quantities vary  and thus we have to deal with more than one  variable. In such situations, we use the concept of partial differentiation.

If u =f(x,y)  ☞ this means ‘u’ is dependent on both ‘x’ & ‘y’. Then,

the partial derivative of the variable u w.r.t x is (∂u/∂x)y  → y is kept constant.
the partial derivative of the variable u w.r.t y is (∂u/∂y)x  → x is kept constant.
Then by fundamental partial differentiation theorem
Total differential of u = du= (∂u/∂x)y dx + (∂u/∂y)x dy.

Notation for partial differentiation:-  The symbol “∂” is used rather than “d” for the differentials.

We come across examples of partial derivatives majorly in thermodynamics,quantum mechanics and in spectroscopic studies.   

I hope with this pellucid explanation one would understand the concept of differentiation and know why is it so helpful to the scientists worldwide. I end my discussion on derivatives with this post.I have just discussed the basics here which are essential for the study of chemistry. In the coming post I shall write about the second part of Calculus – INTEGRATION. Till then,

be a perpetual student of life and keep learning !

Good day !

 

References and Further Reading

1)https://www.tcd.ie/Physics/cpam/teaching/thermodynamics/notes/Thermomath.pdf

2)http://www.columbia.edu/itc/sipa/math/calc_rules_multivar.html

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