# How do you know when to partially differentiate?

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• Zheng_
In summary, the equations for flow acceleration and heat conduction have different forms because they are describing different physical phenomena. The flow acceleration equation considers the changes in velocity of a particle at a fixed location, while the heat conduction equation considers the changes in temperature at any location over time. This is reflected in the different forms of the equations, where the flow acceleration equation includes terms for local and convective acceleration, while the heat conduction equation includes terms for heat storage, net heat flux, and heat generation.
Zheng_
The equation of flow acceleration is

where du/dt is the global acceleration, and ∂u/∂t and u ∂u/∂x are local and convective acceleration respectively.
And the heat conduction equation is

which is heat stored = net heat flux + heat generated.

My question is: why does the ∂T/∂t part in the heat conduction equation is partially differentiated, while the du/dt part in the flow acceleration isn't?

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In the first equation, u is a function of x and t. At any instant (i.e. t constant) there is a velocity at each position x, and a gradient in the velocity field as we consider small changes in x. Likewise, at any location, we can observe particles at that point over time and note how the local velocity changes there.
But if we fix on a particle at some (x,t), not only is it accelerating because of the local acceleration, it may also be accelerating because it is moving to somewhere with a different local velocity. We are interested in how the velocity of this particle changes over time.
It can help to visualise this as a surface, where the horizontal coordinates are x and t, and the height is u. The slope in the x direction shows how much faster or slower nearby particles are moving at an instant, while the slope in the t direction shows how the velocity of particles observed at the fixed location changes with time.
The surface has a tangent plane at this location and time. The slope in any given direction from there can be deduced from the two coordinate slopes.
If we follow a particle from this location, it tracks in a "direction" given by its current speed, u. After time Δt its speed will have changed by ##+\Delta t\frac{\partial u}{\partial t}## by virtue of the acceleration (slope) ##\frac{\partial u}{\partial t}## in the time direction, and by ##\Delta x\frac{\partial u}{\partial x}## by virtue of the slope ##\frac{\partial u}{\partial x}## in the x direction. But we know that ##\Delta x=u\Delta t##, so ##\Delta u=u\Delta t\frac{\partial u}{\partial x}+\Delta t\frac{\partial u}{\partial t}##. Hence the equation you posted.
It is defining u as the velocity of a given particle that is key.

In the temperature equation, we are not fixing on some "packet" of heat that moves around. Instead, ##T=T(\vec x, t)##, where the position and time are independent. There is no ##\frac{dT}{dt}## to be considered. Of course, we could invent a bug that crawls around according to some ##\vec x=\vec x(t)## and ask how it experiences changing temperatures. Then we would have ##\frac{dT_{bug}}{dt}=\frac{\partial T}{\partial t}+\dot{\vec x}.\nabla T##.

Zheng_

## 1. What is partial differentiation?

Partial differentiation is a mathematical concept used in calculus to calculate the rate of change of a function with respect to one of its variables, while holding all other variables constant.

## 2. How do you know when to use partial differentiation?

Partial differentiation is typically used when a function has multiple variables and you want to find the rate of change of the function with respect to one specific variable.

## 3. What is the process of partial differentiation?

The process of partial differentiation involves treating all variables except the one being differentiated as constants, and then differentiating the function with respect to the chosen variable.

## 4. What are some real-life applications of partial differentiation?

Partial differentiation is used in various fields such as physics, engineering, economics, and statistics to analyze and solve problems involving multiple variables and their rates of change.

## 5. Can you give an example of a partial differentiation problem?

An example of a partial differentiation problem could be finding the rate of change of the volume of a cone with respect to its height, while the radius of the base is held constant. This would involve using the formula for the volume of a cone and differentiating it with respect to the height variable.

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