An Easy Application of Total Differential

Calculate the approximate value ofby total differential：
Solution.

The Difference between Delta f and df

As the picture shows that if Δx is small enough(near zero), the Δf and df shall be near.

Total Differential

Consider the function, u(x, y, z) then its total differential is defined:

In here, the dx, dy, dz, is a tiny quantity on each axis, x, y, z, and the

is the rates of change of u(x, y, z) that it is caused by the tiny change of x, and

is the rates of change of u(x, y, z) that it is caused by the tiny change of y, and

is the rates of change of u(x, y, z) that it is caused by the tiny change of z.
Therefore, thecan be explained that it is a tiny quantity of change of u(x, y, z) by tiny change of x, andis a tiny quantity of change by tiny change of y, andis a tiny quantity of change by tiny change of z.
So the total differential du is equal to change of direction x and change of direction y and change of direction z.

Partial Differential (to find the derivatives)

Think of the function u(x, y, z) that has three variables, then the one order partial differential can be defined:











Think of using lower order(s) to define higher order(s) partial differential.
Example:Consider the function u(x, y) that has two variables to define second orders partial differential as following equations.

Mathematics is kind of language