Lecture Note
University
Massachusetts Institute of TechnologyCourse
Multivariable CalculusPages
2
Academic year
2023
Sporkz
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Justify the Chain Rule A first attempt at justifying how to get there might be as follows. 𝑑𝑓 𝑑𝑡 = 𝑓 𝑥 𝑑𝑥 𝑑𝑡 + 𝑓 𝑦 𝑑𝑦 𝑑𝑡 + 𝑓 𝑧 𝑑𝑧 𝑑𝑡 So, we have df equals f sub x dx plus f sub y dy plus f sub z dz. 𝑑𝑓 = 𝑓 𝑥 𝑑𝑥 + 𝑓 𝑦 𝑑𝑦 + 𝑓 𝑧 𝑑𝑧 We know that if x is a function of t, then dx/dt is equal to x prime of t dt: 𝑑𝑥 = 𝑥 𝐼 𝑡 ( )𝑑𝑡 Dy is y prime of t dt: 𝑑𝑦 = 𝑦 𝐼 𝑡 ( )𝑑𝑡 dz is z prime of t dt: 𝑑𝑧 = 𝑧 𝐼 𝑡 ( )𝑑𝑡 So if we plug these into that formula, we'll get that df is f sub x times x prime of t dtplus f sub y y prime of t dt plus f sub z z prime of t dt. 𝑑𝑓 = 𝑓 𝑥 𝑥 𝐼 𝑡 ( )𝑑𝑡 + 𝑓 𝑦 𝑦 𝐼 𝑡 ( )𝑑𝑡 + 𝑓 𝑧 𝑧 𝐼 𝑡 ( )𝑑𝑡 Now, if we have a relation between df and dt, such that df equals something times dt,this means that the rate of change of f with respect to t should equal that samecoefficient. In other words, when we divide by dt, we get the chain rule. Now, well, that kind of works, but it shouldn't be completely satisfactory. So maybeit's not a good idea that we used this differential notation to derive the answer. Infact, it's proved using integration by parts. In this manner, you will not be able to prove the chain rule. At the end of today'slecture, it is likely that we should believe in these objects and what they tell us aboutstrange behavior. However, at this point it is still best for us to be somewhat reluctantto believe what they tell us about unusual properties. The chain rule is not to be applied in this manner. At the end of today's lecture, yes,we should probably believe these strange objects. But so far, we should still be alittle bit reluctant to believe what they tell us about weird things. Δ𝑓≈𝑓 𝑥 Δ𝑥 + 𝑓 𝑦 Δ𝑦 + 𝑓 𝑧 Δ𝑧 Now let's consider changes in x, y, and z as functions of time and see how theychange with time by taking t to be small. Let's divide everything by delta t instead ofjust x. Δ𝑓 Δ𝑡 ≈ 𝑓 𝑥 Δ𝑥+𝑓 𝑦 Δ𝑦+𝑓 𝑧 Δ𝑧 Δ𝑡 Here we are dividing numbers. We don't really know what it means to dividedifferentials, but we do know how to divide numbers. And now if I take delta t very
small, then this item tends to the derivative df dt. Remember that the definition of dfdt is the limit of this ratio when the time interval delta t tends to 0. 𝑤ℎ𝑒𝑛 Δ𝑡→0: Δ𝑓 Δ𝑡 → 𝑑𝑓 𝑑𝑡 The derivation of the derivative may be demonstrated by considering the following: Ifwe choose smaller and smaller values for delta t, then the ratios of the numbersinvolved will tend toward some value. That value is the derivative. Similarly, if wetake dx/dt, where delta t is very small, then this ratio will tend to dx/dt. Δ𝑥 Δ𝑡 → 𝑑𝑥 𝑑𝑡 In particular, if we take the limit as delta t approaches 0, we get df dt on one side. Onthe other side, we get f sub x dx dt plus f sub y dy dt plus f sub z dz dt. 𝑓 𝑥 𝑑𝑥 𝑑𝑡 + 𝑓 𝑦 𝑑𝑦 𝑑𝑡 + 𝑓 𝑧 𝑑𝑧 𝑑𝑡 The approximation becomes better and better as the value of delta t approacheszero. Remember that when we write approximately equal, it means that it's not quitethe same. So when we take a limit as delta t approaches zero, eventually we get anequality.
Justify the Chain Rule
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