Why is it important to specify constant variables in a partial derivative?

[loom91]

Hi,

What does it mean to put a partial derivative in first brackets and put a right subscript to it of another variable?

$$(\frac {\partial Y} {\partial Y})_T$$

Thanks.

Molu

 

[nazzard]

Hello Molu,

it is used to point out which variable should be treated as a constant while calculating the partial derivative.

One example from thermodynamics:

One relation for temperature T and energy E as a function of entropy S and volume V would be:

$$\left( \frac {\partial E(S,V)} {\partial S} \right)=\left( \frac {\partial E} {\partial S} \right)_V=T$$

You can have a look at the following website to see more examples:

http://en.wikipedia.org/wiki/Thermodynamic_potentials

Regards,

nazzard

 

[loom91]

But since a partial derivative indicates that all variables except the variable of partial differentiation are to be kept constant, why is it necessary to specify which one is kept constant?

 

[neutrino]

But since a partial derivative indicates that all variables except the variable of partial differentiation are to be kept constant, why is it necessary to specify which one is kept constant?

It may not always be the case.

Here's an example from Boas...

Let $$z = x^2 - y^2 <br /> <br /> x = r\cos\theta <br /> y = r\sin\theta<br /> <br /> z = 2x^2 - x^2 - y^2<br /> = 2x^2 - r^2 (1)<br /> <br /> z = x^2 + y^2 - 2y^2<br /> = r^2 - y^2 (2)<br /> <br /> z = r^2\cos^2\theta - r^2\sin^2\theta (3)$$

Now calculate $$\left(\frac{\partial z}{\partial r}\right)_x (1) , \left(\frac{\partial z}{\partial r}\right)_y (2) , \left(\frac{\partial z}{\partial r}\right)_\theta (3).$$

 

[neutrino]

Off-topic: Why am I not seeing breaks between the equations? There are one or two lines between them in the code.

 

[Parlyne]

Off-topic: Why am I not seeing breaks between the equations? There are one or two lines between them in the code.

TeX ignores white space in equations unless you tell it to do otherwise.

If you want to introduce a space, you need to type "\ " instead of just " ". To put in a line break, "\\" should work.

Or, you could just use multiple TeX environments.

 

[dextercioby]

But since a partial derivative indicates that all variables except the variable of partial differentiation are to be kept constant, why is it necessary to specify which one is kept constant?

Because usually the functional dependence of the differentiated object is not specified by putting it in round brackets at the right of the object. Functional dependence in thermodynamics is essential, that's why it always matters what variable are to be kept constant.

Daniel.