Vector in Coordinate System

Before you see this post,

see

Eigenvalue in Matrix

Diagonalizable

Total differential

Unit Vector

pic. 1

Let formulas be supposed like that pic. 1.

pic. 2

We can express the vector v like pic. 2.

pic. 3

We can find relationship 

between total differentials in Cartesian and general coordinate system

in pic. 2 and pic. 3.

Total differential

Before you see this post,

see

Derivative

A Property of Limit of A Function : f(x)g(x)

[pic 1]

Δf can be notated like [pic 1].

[pic 2]

We can continuously notate the function like [pic 2].

[pic 3]

Δf can be notated like [pic 3].

[pic 4]

Limit of Δf can be written to df and notated like [pic 4].

[pic 5]

[pic 6]

Let's use definition of partial differential in [pic 4]

and

a property of limit of a function like [pic 6].

See

A Property of Limit of A Function : f(x)g(x)

[pic 7]

We use definition in [pic 5] and a property in [pic 6] at a formula in [pic 4],

we can make a formula in [pic 7].

That is the formula of total differential.