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as I was trying to teach myself financial mathematics, I came across this topic on transforming black scholes pde to a heat equation. I had the exat same question as this post Black Scholes to Heat Equation - Substitution. The answer in that post answered the question perfectly well. However, I could not understand 2 parts.

Firsy is: Why was the total derivative used in the the transformation. When should we use total derivatives instead of just partial derivatives? Even the sign is written in partial derivatives?

For example, I do not understand why couldn't I just use $\frac{\partial U}{\partial \zeta} = \frac{\partial U}{\partial x} \frac{\partial x}{\partial \zeta}$ and then substitute in?

This approach seems to arrive for the same result for $\frac{\partial U}{\partial \zeta}$. But it arrives at different result for $\frac{\partial U}{\partial \tau}$, so I presume it is very wrong. I don't know what is wrong with it.

The second one is why would we need to take derivatives w.r.t $\tau$ at all. Since $\tau$ is just equal to $\tau$, i.e. $\tau = \tau$, what is the intuition behind $\frac{\partial U}{\partial \tau}$ changes? Why are we not just "copying down to the next line" $\frac{\partial U}{\partial \tau}$?

Thank you for any helps in advance. I hope I formulated my question clearly. I am sorry I tried to comment on the original post but I couldn't as I am a new joiner to the forum, so I had to duplicate the question a bit.

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If I understand your questions correctly, $\xi$ is a function of $\tau$ so you need to apply chain rule. Therefore, the rate of change of $U$ wrt $\xi$ also depends on the rate of change wrt $\tau$. Read more on the chain rule wikipedia page

To your 2nd question, we have done a change of variable of $\xi \to x$. But our $\xi$ is also dependent on $\tau$, that is $\xi(\tau)$.

A lot of your confusion can be solved if you write your variables with their dependencies. So here, $U(\xi(\tau),\tau)$.

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  • $\begingroup$ Thank you very much for the help. I am still not so sure about the second question. To try to sum up, if we have change a variable, I need to also calculate the total derivatives of all the variables it depends on and change them as well? So, for example, in this case, only because we changed $\xi$, so we need to change $\tau$ as well only because $\xi$ depends on $\tau$, tho $\tau$ is not directlt changed? Thank you very much for your patience. $\endgroup$
    – David
    Oct 24, 2023 at 11:18
  • $\begingroup$ @David because $\xi$ changed and $U$ is dependent on $\xi$. When you differentiate $U$ wrt $\tau$, you still need to consider $\xi$ because your $\xi$ is dependent on $\tau$. Just take the example $U(\xi (\tau), \tau)) = \xi \tau$, with $\xi = 3c \tau^2$. Differentiate $U$ wrt $\tau$. Then now change $\xi$ to something like $4m\tau^5$. Your derivative of $U$ wrt $\tau$ will be different. If you look at this en.wikipedia.org/wiki/Chain_rule#Example And then change the x or y function to something different, the derivative wrt $t$ will be different even though we didn't change it $\endgroup$ Oct 25, 2023 at 22:42
  • $\begingroup$ @David If you look at the definition of chain rule, you need to "expand" all the dependencies of U. $\endgroup$ Oct 25, 2023 at 22:47
  • $\begingroup$ Thank you for your response again. They are very helpful. So, in terms of when not to. We should only use "PDE only" if there is no interdependencies between variables. If I was correct. $\endgroup$
    – David
    Oct 26, 2023 at 9:02

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