Undergrad Symbol used in total differential and small errors

[goggles31]

Mod note: Thread title changed from "Sign used in total differential and small errors" to "Symbol used in total differential and small errors"
My notes say that z=f(x,y) can be defined as the sum of the changes with respect to x,y respectively. The formula is:

dz = ∂f/∂x * dz + ∂f/∂y * dy

Whereas in small errors, the formula is:

deltaf = ∂f/∂x * deltax + ∂f/∂y * deltay

Shouldn't the sign used be ∂f since it is taken with respect to more than one variable?

 

[Mark44]

My notes say that z=f(x,y) can be defined as the sum of the changes with respect to x,y respectively. The formula is:

dz = ∂f/∂x * dz + ∂f/∂y * dy

Both the text of your notes and the formula above are incorrect. "z = f(x, y)" is not defined as a sum. The differential of z, dz, is defined as a sum. The formula you wrote has a typo, and should be dz = ∂f/∂x * dx + ∂f/∂y * dy or df = ∂f/∂x * dx + ∂f/∂y * dy.

Whereas in small errors, the formula is:

deltaf = ∂f/∂x * deltax + ∂f/∂y * deltay

Shouldn't the sign used be ∂f since it is taken with respect to more than one variable?

No. The actual change in f is $\Delta f$, which is only approximately equal to $\frac{\partial f}{\partial x} \Delta x + \frac{\partial f}{\partial y} \Delta y$.

∂f by itself has no meaning here, but df does. In the calculation of error, you use df to approximate the actual change in function value, $\Delta f$.

If you know $\Delta x$ and $\Delta y$, then $\Delta z \approx dz = \frac{\partial z}{\partial x} \Delta x + \frac{\partial z}{\partial y} \Delta y$.

 

[HallsofIvy]

Mod note: Thread title changed from "Sign used in total differential and small errors" to "Symbol used in total differential and small errors"
My notes say that z=f(x,y) can be defined as the sum of the changes with respect to x,y respectively.

Surely you meant to say that dz, not z, can be defined that way.