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triple - So a bunch of people have been asking me why...

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Unformatted text preview: So a bunch of people have been asking me why they’re getting: ∂z ∂x y ∂x ∂y z ∂y ∂z =1 x I probably said in class that one could do the following: ∂x ∂y = z ∂x ∂z ∂z ∂y y x The problem here is that there’s nothing on the right hand side that says that we’re actually holding z constant. When I had derived the Euler chain rule I did the following: ∂z ∂z dz = dx + dy = 0 ∂x y ∂y x We can state that z is unchanging by setting dz = 0. Then we can move stuﬀ around. ∂z ∂x dx =− dy ∂z ∂y dx = − y x ∂y ∂z ∂z ∂y = y dy x ∂x ∂y z Since there is no z and we are holding it constant manually, (∂x/∂y )z = dx/dy . As you can see, now there’s a −1 present. Crazy huh? 1 ...
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