Question

multivariable differential question, check it out and let me know.

Answer 1

the total differential of a function f(x,y) = dz = partial of f(wrt x) dx +partial of f(wrt y) dy. (wrt is with respect to ). This is a case where I only have the differential given in that sort of form. Trying to get back to z, could I integrate \[\int\limits_{}^{} dz = \int\limits_{}^{} f _{x }dx +\int\limits_{}^{} f _{y}dy\] and get\[ z=x*f_{x} +y* f_{y}\]. Mainly I guess the question is how do the partials behave in such a circumstance. Do they not operate as such after integrating, and if they do, where could I go from here in terms of getting away from the partial derivatives, and the initial total differential.

Answer 2

\[\delta z = \frac{\delta z}{\delta x}dx + \frac{\delta z}{\delta y}dy\]

right?

Answer 3

that should be a 'dz' in the front not a 'delta z'

Answer 4

not according to the chapter on differentials. The del z would still be the partial expression, the total is delta z and then partials next to delta x, and delta y, which is why I'm curious. I know that I could get back to the full funtion by solving within the partials, but dealing with the total differential expressed with the partials and the deltas along with them is an interesting thing especially in just theory and symbology. The complete differential is definitely the sum mentioned, but dealing with integral manipulation raises the question. It's only because messing with exact differential equations, it even came up.