where   is a function of   variables.

Example 1

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where  .

Example 2

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Given   find  

Begin with the definition of the total derivative:  . Notice that in order to continue, we need to calculate   and  

Plugging the results into the definition,  , we find that  

 

Because   can't be negative,  .

 

Tetration and beyond

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  •  
     

The derivative of a polynomial,

 ,

can be defined as

 .

If we use the standard ordered basis

 ,

then

 

can be written as

 ,

and   as

 .

Since

 

satisfies

 

,   represents  .

Wedge product

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General second degree linear ordinary differential equation

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A second degree linear ordinary differential equation is given by

 

One way to solve this is to look for some integrating factor,  , such that

 

Expanding   and setting it equal to

 

 

 

 

 

 

 

 

 

 

 

 

Differential example

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The key to differentials is to think of   as a function from some real number   to itself; and   as a function of some that same real number   to a linear map   Since all linear maps from   to   can be written as a   matrix, we can define   as   and   as

 
 

(As a side note, the value of  , and similarly for all differentials, at   is usually written  .)

Without loss of generality, let's take the function  . Differentiating, we have

 

Since we defined   as   and   as  , we can rewrite the derivative as

 

Multiplying both sides by  , we have

 

And voilà! We can say that for any function  ,