Partial derivaives
This is because as ℎ→0 , ∫ 𝑓(𝑥) 𝑑𝑥 𝛼+ℎ 𝛼
→ℎ𝑓(𝛼)
Now consider a one-dimensional conducting rod lying along the x-axis of from the origin to 𝑥 = 𝐿 . Think about the conservation of thermal energy in a section of the rod 𝛼≤𝑥≤𝛼+ℎ where 𝛼 and h are constants; the geometric setup is illustrated below.
From GCSE physics, we know that:
𝑑𝑚 𝑑𝑥
𝐸 = 𝑚𝑐𝑇 ,
𝜌 =
(𝑓𝑜𝑟 1 𝑑𝑖𝑚𝑒𝑛𝑠𝑖𝑜𝑛)
Where E is thermal energy, m is the mass, c is the specific heat capacity, 𝑇 is the temperature, 𝜌 is the density and 𝑑𝑚 𝑑𝑥 is the ratio of the mass to the distance across which this mass is distributed. We assume that the specific heat capacity and the density of the rod is constant throughout it. Since temperature is different for each of the 𝑑𝑥 , to find the thermal energy across the section 𝛼≤𝑥≤𝛼+ℎ we must sum the values of 𝑇(𝑥, 𝑡) for all of the 𝑑𝑥 segments in this section. Likewise, we let 𝑑𝑥 →0 so that the temperature along the rod is continuous and we make the substitution 𝑑𝑚 = 𝜌𝑑𝑥 . Therefore, the thermal energy in the section of the rod 𝛼≤𝑥≤𝛼+ℎ is given by:
𝛼+ℎ
lim 𝑑𝑥→0
∑ 𝑐𝑇(𝑥, 𝑡)𝜌𝑑𝑥
𝛼
The limit causes the function to change from a discrete summation to a continuous summation and so it is equivalent to:
∫ 𝑐𝑇(𝑥, 𝑡)𝜌𝑑𝑥 𝛼+ℎ 𝛼
We now introduce the heat flux 8 𝑞(𝑥, 𝑡) , which is the rate of flow of thermal energy along the rod in the positive x direction. By definition, the rate at which thermal energy enters the section 𝛼 ≤ 𝑥 ≤ 𝛼+ℎ through its left-hand cross-section at 𝑥 = 𝛼 is 𝑞(𝛼, 𝑡) , while the rate at which thermal energy
8 Dawkins (2003).
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