Maths - Calculus of Different Algebras

So far we have discussed calculus of scalar quantities, however it can be useful to apply the ideas to other algebras, such as:

Differentiation with respect to a scalar

For most algebras adding objects is done by adding the corresponding elements in the object.

For example, in a given algebra, say x is an object consisting of elements x1,x2,x3.. and y is an object consisting of elements y1,y2,y3...

so,

x = [x1,x2,x3...]

y = [y1,y2,y3...]

so adding the objects is done as follows,

x+y = [x1+y1 , x2+y2 , x3+y3 ...]

Differentiation with respect to a scalar is defined as follows, if:

f(x) = [a(x) , b(x) , c(x) , e(x)]

then:

d f(x) / dx = [d(a(x) /dx) , d(b(x)/dx) , d(c(x)/dx) , d(e(x)/dx)]

In other words to differentiate with respect to a scalar, we just differentiate the elements individually. So to give a more specific example if:

f(x) = [xn , sin(x) , tan(x) , ex ]

then:

d f(x) / dx = [n*xn-1 , cos(x) , sec2(x) , ex ]

So this is quite simple, provided that we can differentiate the elements of a vector, we can differentiate the whole object.

Differentiation with respect to non-scalars

We can try following the priciples used already for scalars and use them for other algeras. There is a possible issue: many algebras, such as matrix and quaternion algebras, are not commutative for multipication. Therefore using the notation a divided by b for division can be ambigous. Therefore Leibnitz Notation is not appropiate for some algebras.

 

 

 

 

 

 

 

 

 

 

 

 

 


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