Partial derivaives
Consider a function of two variables 𝑓(𝑥, 𝑦) , since both the first order partial derivatives are functions of 𝑥 and 𝑦 , we can in turn differentiate each in respect to 𝑥 or 𝑦 . This denotes that for a function of two variables, we will have 4 second order partial derivatives.
𝜕 2 𝑓 𝜕𝑥 2
𝜕 2 𝑓 𝜕𝑦𝜕𝑥
𝜕 2 𝑓 𝜕𝑥𝜕𝑦
𝜕 2 𝑓 𝜕𝑦 2
𝑓 𝑥𝑥 =
,
𝑓 𝑥𝑦 =
,
𝑓 𝑦𝑥 =
,
𝑓 𝑦𝑦 =
Below is an example of finding second order partial derivatives for a function.
𝑓(𝑥, 𝑦) = cos(2𝑥) − 𝑥 2 𝑒 5𝑦 +3𝑦 2
5𝑦 , 𝑓
2 𝑒 5𝑦 +6𝑦
𝑓 𝑥 = −2 sin(2𝑥) − 2𝑥𝑒
𝑦 =−5𝑥
5𝑦 ,
5𝑦 ,
5𝑦 , 𝑓
2 𝑒 5𝑦 +6
𝑓 𝑥𝑥 = −4 cos(2𝑥) − 2𝑒
𝑓 𝑥𝑦 =−10𝑥𝑒
𝑓 𝑦𝑥 =−10𝑥𝑒
𝑦𝑦 =−25𝑥
Notice that in this case 𝑓 𝑥𝑦 = 𝑓 𝑦𝑥 . Clairaut’s Theorem 6 tells us that if the second order partial derivatives of a function are continuous, then the order of differentiation is immaterial. For example:
Given 𝑓(𝑥, 𝑦) = 𝑦 3 cos(2𝑥) + 𝑥 3 (𝑦 10 − sin (𝑦 5 )) 17 ,𝑓𝑖𝑛𝑑 𝑓
𝑦𝑥𝑥𝑦𝑥𝑥𝑦
Since there are no logarithms, no non-positive integer indices and so on, there will not be any restrictions on the domains of the variables of the second order partial derivatives and so they are continuous and we can utilize C lairaut’s Theorem.
𝑓 𝑦𝑥𝑥𝑦𝑥𝑥𝑦 = 𝑓 𝑥𝑥𝑥𝑥𝑦𝑦𝑦
3 sin(2𝑥) + 3𝑥 2 (𝑦 10 − sin (𝑦 5 )) 17
𝑓 𝑥 =−2𝑦
3 cos(2𝑥) + 6𝑥(𝑦 10 − sin (𝑦 5 )) 17
𝑓 𝑥𝑥 =−4𝑦
3 sin(2𝑥) + 6(𝑦 10 − sin (𝑦 5 )) 17
𝑓 𝑥𝑥𝑥 =8𝑦
3 cos(2𝑥)
𝑓 𝑥𝑥𝑥𝑥 =16𝑦
2 cos(2𝑥) ,
𝑓 𝑥𝑥𝑥𝑥𝑦 =48𝑦
𝑓 𝑥𝑥𝑥𝑥𝑦𝑦 =96𝑦cos(2𝑥) ,
𝑓 𝑥𝑥𝑥𝑥𝑦𝑦𝑦 = 96 cos(2𝑥)
Deriving the Heat Equation
Firstly, lets introduce the fundamental theory of calculus 7 . Take the function 𝑓(𝑥) graphed below, this theory states that assuming 𝑓(𝑥) is continuous around 𝛼 then:
∫ 𝑓(𝑥) 𝑑𝑥 𝛼+ℎ 𝛼
1 ℎ
lim ℎ→0
= 𝑓(𝛼)
6 Dawkins (2003). 7 Wikipedia n.d.
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