I was reading about exact equations, which I never studied, and that led me to the full differential and differentials in general. None of the calculus nor differential equations books I have cover them; except when it comes to exact equations, and then it only gives the calculation with little to no explanation. For some reason I'm having a difficult time comprehending this topic. Now, I know a differential is an infinitesimal change in that variable, and how to calculate the full differential and full derivative, but I'm having difficulty in some of the meanings.
First, when I take the full differential or full derivative, the answer seems meaningless without numbers. Finally, from my readings, it seems the full differential of a function is approximate to the actual change in the function. Also that the full differential is the true change in the tangent line, plane, surface, etc. Is that true? Thanks.
I think you might be getting confused about partial diffferentiation and its role in calculating the total (full?) derivative. Generally, if you have a function, f = f(x, y) of more than one variable, you can ask how the function changes with just x or just y or both of them together. You can also deal with the situation where y = y(x), i.e., y is a function of x.
A partial derivative of a function is one where the change of the function is taken with respect to (wrt) only one variable with the others held constant. This is written ∂f(x,y)/∂x, where the special symbol "∂" is used to denote that the derivative wrt x-only is calculated and y is considered constant.
Lets say we have a function f(x,y;t) where f is a function of y and y but x and y are also functions of t (the independent variable). If we want to calculate the derivative of f wrt t, we have to take into account this dependence of x and y on t. This is written, using the chain rule for partial derivatives,
df/dt = (∂f/∂t)(dt/dt) + (∂f/∂x)(dx/dt) + (∂f/y)(dy/dt)
= ∂f/∂t + (∂f/∂x)(dx/dt) + (∂f/∂y)(dy/dt), since dt/dt = 1.
Note how the differentials (dx, etc.) cancel, at least superficially in the notation. Also, we can use the total "d" notation instead of the partial "∂" notation since the variables x and y are functions of t-only (don't need ∂).
This total derivative is the actual total change in the function. Expressions like ∂f/∂x are the actual change in the function along the x direction. Your statement about the "change in the tangent line", etc., probably refers to the slope (first derivative) of the function.
The relation to exact equations comes from dealing with 1st order differential equations where
df/dx = (∂f/∂x)(dx/dx) + (∂f/∂y)dy/dx = ∂f/∂x + (∂f/∂y)(dy/dx) which is often written, after multiplying by dx,
df = (∂f/∂x)dx + (∂f/∂y)dy.
Hope this helps. Please ask a follow-up if you need clarification.