I am currently reading "Differential Equatons with Applications" by Ritger and Rose, and I need some clarification about some notation and convention that they are using. I think it all stems from a lack of clarity of the difference between the operator d/dx and the "object" (I dont know what to call it, I dont really know what it is) dx(adsbygoogle = window.adsbygoogle || []).push({});

If I have an equation of the following form

d(f(x,y)) = h(x,y)dx

the approach they then take is to "integrate both sides" and they remain with the following equation

f(x,y) = g(x,y) + c where g(x,y) = ∫h(x,y)dx

the integration on the right is clearly with respect to x, I am not so clear as to what they are doing on the left however. It seems like they are asserting that the integral of the derivative of a function is the function itself. But with respect to what are they integrating it? Is this only with respect to x? If so then the answer is wrong since ∫ d(f(x,y))dx is in general not f(x,y) . Is this some sort of line integral and we are using gradient theorem? If so are we doing the same thing on the other side of the equation? I dont think so, as we are only integrating with respect to x

The other confusion comes from the following, if have the same initial equation as above, d(f(x,y)) = h(x,y)dx is that the same as (d/dx)(f(x,y)) = h(x,y)? If yes, how so? How about if I have something of the following form ∂/∂x (f(x,y)) = h(x,y)?

TIA

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# Confusion about notation and convention regarding differentials as objects and operators

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