where (see also
4-gradient). Sometimes the notation is also used.[1]
Some applications
The multi-index notation allows the extension of many formulae from elementary calculus to the corresponding multi-variable case. Below are some examples. In all the following, (or ), , and (or ).
For an
analytic function in variables one has In fact, for a smooth enough function, we have the similar Taylor expansion where the last term (the remainder) depends on the exact version of Taylor's formula. For instance, for the Cauchy formula (with integral remainder), one gets
For each in , the function only depends on . In the above, each partial differentiation therefore reduces to the corresponding ordinary differentiation . Hence, from equation (1), it follows that vanishes if for at least one in . If this is not the case, i.e., if as multi-indices, then
for each and the theorem follows.
Q.E.D.
^Reed, M.; Simon, B. (1980). Methods of Modern Mathematical Physics: Functional Analysis I (Revised and enlarged ed.). San Diego: Academic Press. p. 319.
ISBN0-12-585050-6.
Saint Raymond, Xavier (1991). Elementary Introduction to the Theory of Pseudodifferential Operators. Chap 1.1 . CRC Press.
ISBN0-8493-7158-9
where (see also
4-gradient). Sometimes the notation is also used.[1]
Some applications
The multi-index notation allows the extension of many formulae from elementary calculus to the corresponding multi-variable case. Below are some examples. In all the following, (or ), , and (or ).
For an
analytic function in variables one has In fact, for a smooth enough function, we have the similar Taylor expansion where the last term (the remainder) depends on the exact version of Taylor's formula. For instance, for the Cauchy formula (with integral remainder), one gets
For each in , the function only depends on . In the above, each partial differentiation therefore reduces to the corresponding ordinary differentiation . Hence, from equation (1), it follows that vanishes if for at least one in . If this is not the case, i.e., if as multi-indices, then
for each and the theorem follows.
Q.E.D.
^Reed, M.; Simon, B. (1980). Methods of Modern Mathematical Physics: Functional Analysis I (Revised and enlarged ed.). San Diego: Academic Press. p. 319.
ISBN0-12-585050-6.
Saint Raymond, Xavier (1991). Elementary Introduction to the Theory of Pseudodifferential Operators. Chap 1.1 . CRC Press.
ISBN0-8493-7158-9