baru
  • baru
Where does the "Total differential: dz= fx dx + fy dy" come from? I'm under the impression that it is from the taylor series... with the intuition that we would need lesser and lesser number of terms in the series to achieve a certain degree of accuracy as the values for the "change in variables (delta x, delta y)" become smaller and smaller, thus If the change in variables were infinitely small..then the series would boil down to just the first terms... am I right? or is the total differential just defined that way?
OCW Scholar - Multivariable Calculus
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katieb
  • katieb
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baru
  • baru
consequently... if an expression such as this occurs: dz = f( )dx + f( )dy + f( )dxdy can we cut out the product of differentials dxdy? i ask because I've come across derivations in other subjects that do exactly this... but the explanation leaves an impression that cutting out products of differentials is a simplification/approximation
phi
  • phi
Have you seen Lecture 11 http://ocw.mit.edu/courses/mathematics/18-02-multivariable-calculus-fall-2007/video-lectures/lecture-11-chain-rule/ It looks like it comes from the tangent plane approximation: at at given point on the curve, the tangent plane at that point approximates the curve. By definition, it is a linear approximation, which precludes terms with higher order deltas. Attached is my interpretation, but I am not a mathematician, so take it for what it's worth.
baru
  • baru
@phi Thanks for that explanation...really helped clear the air about cutting out products of differentials, this is the dervation that is bugging me: its for the speed of sound from "fundamentals of aerodynamics by John D Anderson" this the background rho: density of air a: velocity of wave in air (both are functions of some variables) the law at work is conservation of mass which states: rho * a = constant always objective: find da/d(rho) I've attached a picture of the section of the text(its only four lines) and highlighted the part that bugs me. the author says that the products of differentials are "neglected" , thus suggestive that the resulting expression is an approximation. but the expression can be easily derived this way: \[\rho a = c (constant)\] differentiate with respect to da \[a \frac{ d \rho }{ da } + \rho = 0\] \[a= -\rho da/d \rho\] no approximation involved here...and same result...which means to me that cutting out the product d(rho)da was not an approximation Any idea? or this a question for a mathematician/ this is outside the scope of 18.02 syllabus

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baru
  • baru
@phi a slightly philosophical question i'm not a mathematician either, and my understanding of the "total differential" is still quite vague.. things like this leave me a little unsettled. so the question is: can I consider myself well versed on the topic if i am proficient with its use?
baru
  • baru
would i be wrong if i say\[z = \int\limits_{}^{}fx dx + \int\limits fy dy \] ??
phi
  • phi
for the physics question, they are making a hand-waving argument, but it is not a math book, they just wanted the result and probably did not want to use calculus I don't think many people "understand" differentials. In the lecture, the professor mentions that the textbook gets it wrong! Yes, you can integrate dz as you show, but you do have to be careful because in general both fx and fy are functions of x and y, and as you integrate (for example) \[ \int f_x \ dx\] as x changes, y may also change, hence fx varies. You will see this in Lecture 19 how to handle this.
baru
  • baru
Nice...thanks a lot!

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