What is a total differential? (Geometrically speaking?)

[MathWarrior]

Just got around to learning total differentials in calculus 3. Or total derivative, having a hard time wrapping my mind around what it is geometrically speaking. Can someone explain?

 

[STEMucator]

Just got around to learning total differentials in calculus 3. Or total derivative, having a hard time wrapping my mind around what it is geometrically speaking. Can someone explain?

Lets take a single variable case here, say y=f(x).

So the total differential of f is df = f'(x)dx. You can think of df as a function of two variables, x and dx ( Where dx is also independent here of course ).

Think of an infinitesimally small quantity when you think about dx. So for very very small values of dx you get :

f(x, dx) ≈ f'(x) ( Roughly equal to ).

In a way we cheat the system when we take the limit as h→0 of the definition of the limit. Remember how h is close to 0, but not zero exactly? Think of dx in the same manner. Extremely small, but not zero exactly.

 

[HallsofIvy]

Let f be a differentiable function in x, y, and z (which is stronger than just saying that the partial derivatives exist). Further, suppose x, y, and z are differentiable functions of some parameter t. Then f(t)= f(x(t), y(t), z(t)) describes f values along a specific smooth path in R³. If you like, you can think of it as the trajectory of an object in space.

By the chain rule,
$$\frac{df}{dt}= \frac{\partial f}{\partial x}\frac{df}{dt}+ \frac{\partial f}{\partial y}\frac{dy}{dt}+ \frac{\partial f}{\partial z}\frac{dz}{dt}$$

Since that is an ordinary derivative of a function of a single variable, we can define its "differential":
$$df= \frac{df}{dt} dt= \left(\frac{\partial f}{\partial x}\frac{df}{dt}+ \frac{\partial f}{\partial y}\frac{dy}{dt}+ \frac{\partial f}{\partial z}\frac{dz}{dt}\right)dt$$

Or course, that is the same as
$$df= \frac{df}{dt} dt= \frac{\partial f}{\partial x}\frac{df}{dt}dt+ \frac{\partial f}{\partial y}\frac{dy}{dt}dt+ \frac{\partial f}{\partial z}\frac{dz}{dt}dt$$

$$df= \frac{\partial f}{\partial x}dx+\frac{\partial f}{\partial y}dy+ \frac{\partial f}{\partial z}dz$$
where all mention of "t" has disappeared.