# Partial Derivative multiplied by increment (How can I think of 'the size' of (f.e.) $\partial\,t$ relative to $\delta\,T$?)

#### I'mLearning

##### New member
Let the local temperature change of a moving baloon be

$\delta\,T=\frac{\partial\,T}{\partial\,x}\delta\,x+\frac{\partial\,T}{\partial\,t}\delta\,t$
The use of the delta-symbol implies the idea of an increment and the partial derivative is defined as a limit.

My question is: How can I think of 'the size' of (f.e.) $\partial\,t$ relative to $\delta\,T$?
Can they be compared or is this not possible from a more formal point of view?
Is the delta-symbol used as a symbol for an arbitrary increment?

I apologize if I fail to express myself correctly.

Last edited:
Let the local temperature change of a moving baloon be

$\delta\,T=\frac{\partial\,T}{\partial\,x}\delta\,x+\frac{\partial\,T}{\partial\,t}\delta\,t$
The use of the delta-symbol implies the idea of an increment and the partial derivative is defined as a limit.

My question is: How can I think of 'the size' of (f.e.) $$\partial\,t$$ relative to $$\delta\,T$$?
Can they be compared or is this not possible from a more formal point of view?
Is the delta-symbol used as a symbol for an arbitrary increment?

I apologize if I fail to express myself correctly.
Please show us where this question came from. Is it your own idea, or something from your textbook (or class notes)? If the latter, an image of how it was stated will help. Are you learning the multivariable chain rule, or something else?

In particular, I want to see whether your book actually treats [imath]\partial\,t[/imath] as an entity in itself as you are doing, and how it uses other notation.

Let the local temperature change of a moving baloon be

$\delta\,T=\frac{\partial\,T}{\partial\,x}\delta\,x+\frac{\partial\,T}{\partial\,t}\delta\,t$
The use of the delta-symbol implies the idea of an increment and the partial derivative is defined as a limit.

My question is: How can I think of 'the size' of (f.e.) $\partial\,t$ relative to $\delta\,T$?
Can they be compared or is this not possible from a more formal point of view?
Is the delta-symbol used as a symbol for an arbitrary increment?

I apologize if I fail to express myself correctly.
The way I see it there is no separate entity [imath]\partial t[/imath], only [imath]\frac{\partial T}{\partial t}[/imath], which is a notation for "approximate dependency of 'T' on 't' "