Correct Usage of Partial Derivative Symbols in PDEs

[nizi]

Homework Statement

In the following second-order partial differential equation for $f$ with $x$ and $t$ as independent variables, $a$ and $b$ as constants, and $g$ as a known function with $t$ as the only independent variable, is the mathematically correct notation for the third term on the left-hand side $\frac{ dg }{ dt }$ as below, or $\frac{ \partial g }{ \partial t }$?

Relevant Equations

$$a \frac{ \partial^2 f}{ \partial x^2 } + b \frac{ \partial f }{ \partial t } + \frac{ dg }{ dt } = 0$$

Some may say that $\frac{ \partial g }{ \partial t }$ is correct because it is a term in a partial differential equation, but since $g$ is a one variable function with $t$ only, I think $\frac{ dg }{ dt }$ is correct according to the original usage of the derivative and partial derivative symbol.
The original usage of the partial derivative symbol is to express the rate of change of a multivariable function of two or more variables when all variables except the variable to be used in the differentiation are fixed, and the notation $\frac{ \partial g }{ \partial t }$ implies that $g$ is not a one variable function, which would be somewhat inaccurate.

 

[fresh_42]

Both can be used but $\dfrac{dg}{dt}$ is clearer as we have only $g=g(t).$ It also has historical reasons to write it that way. $\dfrac{\partial g}{\partial t}$ would be misleading because everyone would search for the other variables, e.g. for $a$ or $b$ as parametric variables.

I would only use the partials if I explicitly considered the partial derivatives as a basis of a common vector space where I wanted to express all functions, $\dfrac{\partial }{\partial x_i}f=\dfrac{\partial }{\partial x_i}f(\boldsymbol x,t)\, , \,\dfrac{\partial }{\partial t}f=\dfrac{\partial }{\partial t}f(\boldsymbol x,t)\, , \,\dfrac{\partial }{\partial t}g=\dfrac{\partial }{\partial t}g(t)$, i.e. the vector space of differential forms. But even then it is more likely to consider $\left\{\dfrac{\partial }{\partial x_i}\right\}$ as a basis, and $\dfrac{\partial }{\partial t}g(t)$ as a flow within that space.

Edit: ... that would have been better written as $\dfrac{d}{dt}g(t).$Long story short: $\dfrac{d}{dt} g(t)$ is correct.

 

[nizi]

Thank you for your detailed response, fresh_42-san (I am Japanese, and in Japanese, -san is added after the name as an honorific title.).

> $\frac { \partial g} { \partial t}$ would be misleading because everyone would search for the other variables
I completely agree with you.

By the way, in the following part, isn't $\frac { \partial } { \partial t } g(t)$ a mistake for $\frac {\partial} {\partial t}$?
> $\frac { \partial } { \partial t } g(t)$ as a flow within that space

 

[dextercioby]

Putting dg/dt in the PDE for the unknown "f" is correct but also misleading and to be avoided, for it simply implies that g(t) is also an unknown function (thus its derivative is also unknown). If g was known, its derivative can be calculated and expressed in terms of "t". Misleading because one is led to believe he has to solve a wierdly mixed DE with 2 unknowns, or that someone forgot to put a second (again possibly mixed) DE to come to the known pattern "n equations, n unknowns".

 

[fresh_42]

Thank you for your detailed response, fresh_42-san (I am Japanese, and in Japanese, -san is added after the name as an honorific title.).

> $\frac { \partial g} { \partial t}$ would be misleading because everyone would search for the other variables
I completely agree with you.

By the way, in the following part, isn't $\frac { \partial } { \partial t } g(t)$ a mistake for $\frac {\partial} {\partial t}$?
> $\frac { \partial } { \partial t } g(t)$ as a flow within that space

Corrected. It was late when I wrote it and derivatives make you dizzy. I have counted $10$ different views on a derivative here:
https://www.physicsforums.com/insights/journey-manifold-su2mathbbc-part/
and "slope" wasn't even among them.

I would briefly say:
\begin{align*}
dx_k \quad &\text{differential forms }\\
\dfrac{\partial f}{\partial x_k} \quad &\text{coordinates (or components) of }df\text{ in the vector space of differential forms }\\
\dfrac{d}{dt}g(t)\quad &\text{a flow }\\
\dfrac{\partial }{\partial t}\quad &\text{unneccessary and confusing, because time and space are not connected in this set-up}
\end{align*}

 

[WWGD]

Still, $\partial/\partial t$
Is often used as a basic vector field in Differential Geometry. For a 2D tangent space , e.g, a basis is noted as $\{ \partial/\partial x, \partial/ \partial y \}$
Edit. These are usually the directional derivatives in the " standard directions" , along the x,y directions, often also called $e_1,e_2$, viewed as operators on functions.

 

[nizi]

Thank you all for your kind responses.
I am currently out of the country on a business trip, so please give me some time to respond.
I apologize for the delay in responding to your question.

 

[nizi]

Putting dg/dt in the PDE for the unknown "f" is correct but also misleading and to be avoided, for it simply implies that g(t) is also an unknown function (thus its derivative is also unknown). If g was known, its derivative can be calculated and expressed in terms of "t". Misleading because one is led to believe he has to solve a wierdly mixed DE with 2 unknowns, or that someone forgot to put a second (again possibly mixed) DE to come to the known pattern "n equations, n unknowns".

I apologize for the delay. I sincerely appreciate your kind remarks.
Since $g$ is known, you are saying that I should differentiate by $t$ in advance before notating the equation. Indeed, my equation notation gives the impression that $g$, like $f$, is an unknown function to be obtained from the partial differential equation.
In this case, I modified the equation from the original one to highlight the part I wanted to ask about rather than the partial differential equation itself, which is the subject of my question.
This time, I have modified the equation from the original one to highlight the part I wanted to inquire about, rather than the partial differential equation itself, which is the subject of the question. In fact, I used a generic notation because the original partial differential equation has two patterns for $g$.
In the future, when I encounter similar situations, I'll keep this in mind and describe partial differential equations.

 

[nizi]

Corrected. It was late when I wrote it and derivatives make you dizzy. I have counted $10$ different views on a derivative here:
https://www.physicsforums.com/insights/journey-manifold-su2mathbbc-part/
and "slope" wasn't even among them.

I would briefly say:
\begin{align*}
dx_k \quad &\text{differential forms }\\
\dfrac{\partial f}{\partial x_k} \quad &\text{coordinates (or components) of }df\text{ in the vector space of differential forms }\\
\dfrac{d}{dt}g(t)\quad &\text{a flow }\\
\dfrac{\partial }{\partial t}\quad &\text{unneccessary and confusing, because time and space are not connected in this set-up}
\end{align*}

I apologize for the delay. I sincerely appreciate your kind remarks.
I checked the link, and although I have heard of vector bundle, Lie group, etc., the level of mathematics is too high for me to decipher. I am sorry for the trouble you have gone to in teaching me this. I have dabbled in differential geometry, so I managed to understand the summary at the end of this forum.
However, the world of mathematics is very deep. I never realized that there are as many as 10 different perspectives on a single derivative.
I welcome these posts that expand my world.
I know it will take some time, but I will study them with reference to fresh_42-san's other writings!

 

[nizi]

Still, $\partial/\partial t$
Is often used as a basic vector field in Differential Geometry. For a 2D tangent space , e.g, a basis is noted as $\{ \partial/\partial x, \partial/ \partial y \}$
Edit. These are usually the directional derivatives in the " standard directions" , along the x,y directions, often also called $e_1,e_2$, viewed as operators on functions.

I apologize for the delay. I sincerely appreciate your kind remarks.
$\{ \frac { \partial }{ \partial x }, \frac { \partial }{ \partial y } \}$ is the basis of the tangent space, which I studied in linear algebra.
WWGD-san, like fresh_42-san, has given me another perspective of partial derivatives as a basis.
I would like to start with a review of differential geometry first, and then study the remaining perspectives on partial derivatives presented by fresh_42-san.

 

[pasmith]

Putting dg/dt in the PDE for the unknown "f" is correct but also misleading and to be avoided, for it simply implies that g(t) is also an unknown function (thus its derivative is also unknown).

How would you indicate the derivative of an arbitrary function? It's not "unknown", in the sense that it must be specified before the PDE can be solved rather than being obtained as part of the solution, but nonetheless it is not possible to give its derivative in terms of $t$ before it is specified.