# Question about the definition of a partial derivative

• B
• lerus
In summary, the different partial derivatives that are calculated when a function is differentiated depend on which variables are held fixed.
lerus
I just started to study thermodynamics and very often I see formulas like this:
$$\left( \frac {\partial V} {\partial T} \right)_P$$
explanation of this formula is something similar to:
partial derivative of ##V## with respect to ##T## while ##P## is constant.
But as far as I remember partial derivative is always calculated when all variables except one (##T## in our case) are constant.

Is it correct that:
$$\left( \frac {\partial V} {\partial T} \right)_P$$
the same as
$$\left( \frac {\partial V} {\partial T} \right)$$

Thank you

(∂V∂T)P

lerus said:
Is it correct that:
$$\left( \frac {\partial V} {\partial T} \right)_P$$
the same as
$$\left( \frac {\partial V} {\partial T} \right)$$
Yes, BUT only if ##V(T,P)##.

Since the physical quantity ##V## can be seen as
different functions of different pairs of independent variables,
it's best in thermodynamics to be explicit about what is being held constant.

jbergman and topsquark
For instance, if we have function ## V \left( T, P, X \right) ##

is ## \left( \frac {\partial V} {\partial T} \right)_P \equiv \left( \frac {\partial V} {\partial T} \right) ##

Thank you.

If you don't specify what ##V## depends on and what is held fixed,
I can't compute any partial derivatives.

vanhees71 and PeroK
For instance, we have function ## V \left( T, P, X \right) ##
When we calculate ## \left( \frac {\partial V} {\partial T} \right) ##
we change ##T## and keep ##P## and ##X## constant
When we calculate ## \left( \frac {\partial V} {\partial T} \right)_P ## we do the same
Then why do we need ##_P## ?

Thank you

From Schroeder's Thermal Physics p. 31
Problem 1.45. As an illustration of why it matters which variables you hold fixed
when taking partial derivatives, consider the following mathematical example.
Let ##w = xy## and ##x = yz##.
(a) Write ##w## purely in terms of ##x## and ##z##, and then purely in terms of ##y## and ##z##.

(b) Compute the partial derivatives
$$\left(\frac{\partial w}{\partial x}\right)_y \qquad\mbox{and}\qquad \left(\frac{\partial w}{\partial x}\right)_z$$
and show that they are not equal.
(Hint: To compute ##\left(\frac{\partial w}{\partial x}\right)_y## use a formula for ##w## in terms of ##x## and ##y##, not ##z## .
Similarly, compute ##\left(\frac{\partial w}{\partial x}\right)_z## from a formula for ##w## in terms of only ##x## and ##z##.)

(c) Compute the other four partial derivatives
(two each with respect to ##y## and ##z##),
and show that it matters which variable is held fixed.

jbergman, vanhees71, jasonRF and 2 others
Thanks a lot for example,
I think I understand it better now
If variables ##x, y, z## were independent then ## \left( \frac {\partial w} {\partial x} \right)_y \equiv \left( \frac {\partial w} {\partial x} \right) _z##
but if ##x, y, z## are not independent, then
## \left( \frac {\partial w} {\partial x}\right)_y \neq \left( \frac {\partial w} {\partial x}\right)_z##

Thank you

vanhees71
lerus said:
Thanks a lot for example,
I think I understand it better now
If variables ##x, y, z## were independent then ## \left( \frac {\partial w} {\partial x} \right)_y \equiv \left( \frac {\partial w} {\partial x} \right) _z##
but if ##x, y, z## are not independent, then
## \left( \frac {\partial w} {\partial x}\right)_y \neq \left( \frac {\partial w} {\partial x}\right)_z##

Thank you
Well, the root of the problem is general sloppiness in using the same symbol for different functions. The more mathematically precise solution, in this example, would be to define:
$$w(x, y) = xy, \ \text{and} \ \bar w(y, z) = y^2z$$And then it's clear what function is being differentiated.

jbergman, vanhees71 and jasonRF
PS and then it's obvious and trivial that not all functions have the same partial derivatives!

jbergman and vanhees71
https://bridge.math.oregonstate.edu/papers/bridge.pdf
Bridging the Gap between Mathematics and the Physical Sciences
Tevian Dray and Corinne Manogue

Dray & Manogue said:
2 An Example

Here’s our favorite example: Suppose ##T(x, y) = k(x^2 + y^2 )##.
What is ##T(r, θ)##?
We often ask this question of mathematicians and other scientists.
Some mathematicians say “##k(r^2 + θ^2 )##”. Many mathematicians refuse to answer, claiming that the question is ambiguous. Everyone else, including some mathematicians, says “##kr^2##”. One colleague, who holds a split appointment in mathematics and physics, simply laughed, then asked which hat he should wear when answering the question. What’s going on here?
...

And yes, a physicist really will write ##T(x, y) = k(x^2 + y^2)## for, say, the temperature on a
rectangular metal slab, and ##T(r, θ) = kr^2## for the same temperature in polar coordinates,
even though the mathematician would argue that the symbol ##T## is being used for two different
functions. This is not sloppy mathematics on the part of the physicist; it’s a different language.
##T## is the temperature, a physical quantity which is a function of position;
the letters which follow merely indicate which coordinate system one is using to label the position.
This can be rigorously translated into the differential geometer’s notion of a scalar field,

So not only do other scientists speak a different language, they use the same vocabulary!
...

vanhees71
robphy said:
https://bridge.math.oregonstate.edu/papers/bridge.pdf
Bridging the Gap between Mathematics and the Physical Sciences
Tevian Dray and Corinne Manogue
I feel that needs that to be made clear for the student's benefit. Ultimately, you may know when you mean by ##T## in every context, but it's a source of confusion if the student is left to guess what you mean.

In this case, that ##T## is a physical quantity with different functional identities should be made clear from the outset. You can't do physics without calculus and calculus is carried out on functions. You need both physical quantities and functions.

Another good example is the "total" derivative. Physics texts often make a big play about the difference between a total derivative and a partial derivative. The total derivative, however, is simply the usual derivative of a plain old single-variable function.

The danger is that you end up with a sort of mystique about these things. Whereas, if the student can ultimately reduce things to the basics of calculus, then he/she is able to disentangle things for themselves and avoid misconceptions.

jbergman and vanhees71
lerus said:
Thanks a lot for example,
I think I understand it better now
If variables ##x, y, z## were independent then ## \left( \frac {\partial w} {\partial x} \right)_y \equiv \left( \frac {\partial w} {\partial x} \right) _z##
but if ##x, y, z## are not independent, then
## \left( \frac {\partial w} {\partial x}\right)_y \neq \left( \frac {\partial w} {\partial x}\right)_z##

Thank you
If you have the time and inclination, you may be interested in my insight on demystifying the chain rule(!):

https://www.physicsforums.com/insights/demystifying-chain-rule-calculus/

lerus
PeroK said:
I feel that needs that to be made clear for the student's benefit. Ultimately, you may know when you mean by ##T## in every context, but it's a source of confusion if the student is left to guess what you mean.
I'd also say the mathematicians are here too soft. It's indeed a very bad habit of physicists to use the same symbol for different functions, distinguishing the functions only by the naming of the arguments. This can lead to serious confusion. What drives my nuts, e.g., is to use the same symbol for a function and its Fourier transform.

In thermodynamics, it's on the other hand a somewhat different story, because the various quantities have a fixed meaning, no matter as functions of which other variables you use them. Then it's, of course, important to list all (!) the independent variables to be held fixed when differenting with respect to the only other independent variable. So there is some justification for this quite confusing notation, although I think it contributes a lot to the difficulties everybody has with thermodynamics.
PeroK said:
In this case, that ##T## is a physical quantity with different functional identities should be made clear from the outset. You can't do physics without calculus and calculus is carried out on functions. You need both physical quantities and functions.

Another good example is the "total" derivative. Physics texts often make a big play about the difference between a total derivative and a partial derivative. The total derivative, however, is simply the usual derivative of a plain old single-variable function.

The danger is that you end up with a sort of mystique about these things. Whereas, if the student can ultimately reduce things to the basics of calculus, then he/she is able to disentangle things for themselves and avoid misconceptions.
I couldn't agree more ;-).

PeroK

## What is the definition of a partial derivative?

A partial derivative is a mathematical concept that measures the rate of change of a function with respect to one of its variables while holding all other variables constant. It is denoted by ∂ (the partial symbol) and is often used in multivariable calculus.

## How is a partial derivative different from a regular derivative?

A regular derivative measures the rate of change of a function with respect to one variable, while a partial derivative measures the rate of change with respect to a specific variable while holding all other variables constant. In other words, a regular derivative is a special case of a partial derivative.

## What is the purpose of using partial derivatives?

Partial derivatives are useful in many areas of mathematics and science, including physics, economics, and engineering. They allow us to analyze how changing one variable affects the overall behavior of a function, and can be used to optimize functions and solve complex equations.

## How do you calculate a partial derivative?

To calculate a partial derivative, you first need to determine which variable you are taking the derivative with respect to. Then, you treat all other variables as constants and use the standard rules of differentiation to find the derivative. For example, if the function is f(x,y) and you are taking the partial derivative with respect to x, you would treat y as a constant and differentiate f(x,y) with respect to x.

## What are some real-world applications of partial derivatives?

Partial derivatives are used in many fields, including physics, economics, and engineering. For example, in physics, they are used to study the motion of objects in multiple dimensions, while in economics, they can be used to analyze the relationship between different variables in a market. In engineering, partial derivatives are used to optimize designs and analyze the behavior of complex systems.

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