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User:Foxjwill/Math stuff

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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 ,