Let be an open bounded subset of , assume we have a finite open covering of .

Consider the partition of unity subordinated: and where .

How does it follow that ?

Thanks!

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- Sep 10th 2013, 10:27 AMJohnyboyDel Operator
Let be an open bounded subset of , assume we have a finite open covering of .

Consider the partition of unity subordinated: and where .

How does it follow that ?

Thanks! - Sep 10th 2013, 10:47 PMzzephodRe: Del Operator

then as and the result follows.

Though what your first sentence has to do with the rest of the question escapes me.

. - Sep 11th 2013, 03:48 AMJohnyboyRe: Del Operator
Thanks for response, for every open cover there exists a partition of unity.

- Sep 11th 2013, 07:57 AMzzephodRe: Del Operator
- Sep 11th 2013, 01:19 PMJohnyboyRe: Del Operator
Okay fair enough. I have another short question:

Since are real-valued functions, are vector fields according to my understanding of how the operator works.

In the book I am working from I am given that for it follows that has compact support and it follows then that

each has compact support? But is a vector field, does the notion of compact support extend to vector fields?

If so how?

Secondly it is stated that , but too is a vector field.

My understanding is that implies where is measurable and is bounded a.e..

Am I missing something? Thanks.