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...
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)]
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 ]
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 for division can be ambigous. Therefore Leibnitz Notation is not appropiate for some algebras.