Total Differential

from 東大1S熱力学 Total Differential

• Mathematical Understanding using Taylor Expansion
  • $df=\frac{\partial f}{\partial x}dx+\frac{\partial f}{\partial y}dy+\frac{\partial f}{\partial z}dz$
    • I don’t understand this (blu3mo)
    • $df:=f(x+b,y+b,z+b)-f(x,y,z)$
      • I understand this (blu3mo)
    • It seems that this is the Taylor Expansion using the notation of partial derivatives.
      • $f(x+dx)=f(x)+\frac{df}{dx}dx$ (only the first and second terms)
    • Taylor Expansion of Multivariable Functions - EMAN’s Mathematical Physics
      • Oh, I see that Taylor Expansion can also be done for multivariable functions.
      • I didn’t know that.
      • It seems like I can understand why this is the case from the specific image below (blu3mo)
• Understanding through a Concrete Image
  • https://www.youtube.com/watch?v=ChoArVJnSjQ
    • $df$ is simply a small change, and it is only when you divide it by a small amount that you can obtain slopes and so on.
      • I misunderstood (blu3mo)
      • Wrong: $df:={\lim }_{b\to 0}\frac{f(x+b,y+b,z+b)-f(x,y,z)}{b}$
      • Correct: $df:=f(x+d,y+d,z+d)-f(x,y,z)$
    • The calculation is basically taking Partial Differentiation for each variable and adding them together.
    • Ahhh, I got the image (blu3mo)(blu3mo)(blu3mo)
      • By taking the partial derivative of x and multiplying it by the change in x (dx), you can obtain the change in f in the x direction.
        • Similarly for other variables.
      • By adding them together, you can obtain the total change in f obtained from the slopes in all directions (blu3mo)(blu3mo)
      • I see (blu3mo)(blu3mo)(blu3mo), I feel like I also understood the meaning of dx and dy together.
    • I also recommend the explanation in Total Differential - EMAN’s Analytical Mechanics (takker)
      • It explains why simply adding the partial differentials gives the total differential through equation transformations (blu3mo)
        • I see 🙏
  • Taking the dot product of (dx, dy, dz) and Differential Operator indeed gives the total differential (blu3mo)