Mackey space - Revision history

en>Monkbot: /* References */Task 4: Fix CS1 deprecated coauthor parameter errors

2014-07-13T00:10:09Z

References: Task 4: Fix CS1 deprecated coauthor parameter errors

← Older revision		Revision as of 02:10, 13 July 2014
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	Huttons Real Property Group is likely one of the quickest growing actual estate companies with over 3000 actual estate consultants and over a decade of project marketing and consulting expertise. Based in 2002, we are primarily based in Singapore with partners at present in Malaysia (Huttons OneWorld Snd Bhd) and Philippines (Huttons MHT Co Inc). We are nonetheless growing rapidly and fast expanding internationally, making it easier for both our purchasers and patrons to cross-purchase and promote property domestically and overseas.<br><br>Whitehaven @ Pasir Panjang – A boutique development effectively nestled peacefully in serene Pasir Panjang personal property presenting 120 uncommon freehold private flats tastefully designed by the famend Ong & Ong Architect. Only a quick drive away from ~~Science Park and NUS Campus~~, ~~Jade Residences, a up~~ to date Freehold condominium which supplies quality lifestyle with excellent amenities and conveniences proper at its door steps. Its fashionable linear architectural type promotes peace and tranquility residing nestled within the D19 non-public housing enclave. Rising workplace sector leads real estate market performance, ~~while prime retail and enterprise park segments average and residential sector continues in decline Global Market Perspectives - 1st Quarter 2014<br><br>This website is visited by many Singapore~~ property hunters EVERYDAY! My objective is to attract native & overseas consumers who're taken with buying Singapore properties & real estates. I'm committed to providing quality service to all my purchasers. And with my local & abroad database, I am at all times eager to tell & replace my consumers on the most recent and finest purchase properties and real estates in Singapore. Simply simply name me or refill my ( Selling / Leasing Enquiry ) form when you are able to put up your property in the marketplace.Your property will obtain its personal web web page which incorporates full property data and images, hyperlinks to [http://~~www~~.~~hjsq.com~~/~~profile/ChelseBevi Www.hjsq.com~~] ~~maps, community information~~ and ~~/ or streaming video & Digital Tours~~.<br><br>Only Singapore citizens and accepted persons can buy Landed 'residential property'as defined in the Residential Properties Act. Foreigners are eligible to purchaseunits in condominiums or residences which are not landed dwelling homes. Foreignerswho wish to buy landed property in Singapore must first search the approval ofthe Controller of Residential Property. Your Lawyer's Position in a Property Buy The biggest errors Singapore property investors and novices make Secrets of Singapore Property Gurus evaluation by Way of life magazine The Laurels is a freehold house development at Cairnhill Road, Singapore (District 9). Property developer of recent apatments. 107 Tampines Highway, Singapore 535129. Search HDB Flats For Sale by Estate Industrial Properties<br><br>To be taught more about TheLibrary collections of 1 bedroom and a pair of bedroom house floor plansplease sign up for a VIP Private Presentation or attend considered one of ourweekend London Property Speak offered by UK Property Specialist. It's free, and you will mechanically obtain property market updates, the newest real property developments, new pointers and insurance policies, and lots of more. Discover out extra Freehold Condominium near NUS and Biomedical Hub Rare Landed Housing Estate in Serangoon Garden The Authorities runs an everyday programme of releasing State lands on the market to the non-~~public sector for residential, office, retail, industrial~~ and other developments in keeping with market demand. Clothes Business For Sale, Ladies Put on Enterprise For Sale, Wholesale Business For Sale<br><br>Food Enterprise For Sale, Transportation Business For Sale Training & School Related Enterprise For Sale, Training Enterprise For Sale Training Enterprise For Sale, Swimming Pool Enterprise For Sale All Manufacturing Enterprise For Sale, Other Technical Services Business For Sale Displaying 1 to 45 Businesses For Sale of 119 D19) Hougang / Punggol / Sengkang Rental Widespread Room with Aircon D09) Orchard / River Valley Condo Common Room A property agent helped them sell it to a family of 4 who've moved in. F.A.Q for individuals who are planning to search for JOB right here in SG. Scorching Property! Listed here are the top new properties for this week. Click right here for extra! Property Barons Pte Ltd On-line actual estate market place. A one-cease service from hunting for the property to monetary arrangements.		Zoologist Stephan from Grand Valley, likes to spend some time kite boarding, property developers [http://Westwoodchristian.org/?option=com_k2&view=itemlist&task=user&id=69149 condo for sale In singapore] singapore and base jumping. Discovered some amazing locales having spent 9 days at Mont-Saint-Michel and its Bay.

en>Xqbot: Robot: Adding missing tag

2014-02-07T08:03:52Z

Robot: Adding missing <references /> tag

← Older revision		Revision as of 10:03, 7 February 2014
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	~~{{Unreferenced\|date=December 2007}}~~		Huttons Real Property Group is likely one of the quickest growing actual estate companies with over 3000 actual estate consultants and over a decade of project marketing and consulting expertise. Based in 2002, we are primarily based in Singapore with partners at present in Malaysia (Huttons OneWorld Snd Bhd) and Philippines (Huttons MHT Co Inc). We are nonetheless growing rapidly and fast expanding internationally, making it easier for both our purchasers and patrons to cross-purchase and promote property domestically and overseas.<br><br>Whitehaven @ Pasir Panjang – A boutique development effectively nestled peacefully in serene Pasir Panjang personal property presenting 120 uncommon freehold private flats tastefully designed by the famend Ong & Ong Architect. Only a quick drive away from Science Park and NUS Campus, Jade Residences, a up to date Freehold condominium which supplies quality lifestyle with excellent amenities and conveniences proper at its door steps. Its fashionable linear architectural type promotes peace and tranquility residing nestled within the D19 non-public housing enclave. Rising workplace sector leads real estate market performance, while prime retail and enterprise park segments average and residential sector continues in decline Global Market Perspectives - 1st Quarter 2014<br><br>This website is visited by many Singapore property hunters EVERYDAY! My objective is to attract native & overseas consumers who're taken with buying Singapore properties & real estates. I'm committed to providing quality service to all my purchasers. And with my local & abroad database, I am at all times eager to tell & replace my consumers on the most recent and finest purchase properties and real estates in Singapore. Simply simply name me or refill my ( Selling / Leasing Enquiry ) form when you are able to put up your property in the marketplace.Your property will obtain its personal web web page which incorporates full property data and images, hyperlinks to [http://www.hjsq.com/profile/ChelseBevi Www.hjsq.com] maps, community information and / or streaming video & Digital Tours.<br><br>Only Singapore citizens and accepted persons can buy Landed 'residential property'as defined in the Residential Properties Act. Foreigners are eligible to purchaseunits in condominiums or residences which are not landed dwelling homes. Foreignerswho wish to buy landed property in Singapore must first search the approval ofthe Controller of Residential Property. Your Lawyer's Position in a Property Buy The biggest errors Singapore property investors and novices make Secrets of Singapore Property Gurus evaluation by Way of life magazine The Laurels is a freehold house development at Cairnhill Road, Singapore (District 9). Property developer of recent apatments. 107 Tampines Highway, Singapore 535129. Search HDB Flats For Sale by Estate Industrial Properties<br><br>To be taught more about TheLibrary collections of 1 bedroom and a pair of bedroom house floor plansplease sign up for a VIP Private Presentation or attend considered one of ourweekend London Property Speak offered by UK Property Specialist. It's free, and you will mechanically obtain property market updates, the newest real property developments, new pointers and insurance policies, and lots of more. Discover out extra Freehold Condominium near NUS and Biomedical Hub Rare Landed Housing Estate in Serangoon Garden The Authorities runs an everyday programme of releasing State lands on the market to the non-public sector for residential, office, retail, industrial and other developments in keeping with market demand. Clothes Business For Sale, Ladies Put on Enterprise For Sale, Wholesale Business For Sale<br><br>Food Enterprise For Sale, Transportation Business For Sale Training & School Related Enterprise For Sale, Training Enterprise For Sale Training Enterprise For Sale, Swimming Pool Enterprise For Sale All Manufacturing Enterprise For Sale, Other Technical Services Business For Sale Displaying 1 to 45 Businesses For Sale of 119 D19) Hougang / Punggol / Sengkang Rental Widespread Room with Aircon D09) Orchard / River Valley Condo Common Room A property agent helped them sell it to a family of 4 who've moved in. F.A.Q for individuals who are planning to search for JOB right here in SG. Scorching Property! Listed here are the top new properties for this week. Click right here for extra! Property Barons Pte Ltd On-line actual estate market place. A one-cease service from hunting for the property to monetary arrangements.

	~~{{main\|Laplace operator}}~~

	~~==−div is adjoint to d==~~
	~~The claim is made that −div~~ is ~~adjoint to ''d'':~~

	~~:<math>\int_M df(X) \;\omega = - \int_M f \, \operatorname{div} X \;\omega </math>~~

	~~Proof~~ of the ~~above statement:~~
	~~:<math>\int_M (f\mathrm{div}~~(X) ~~+ X~~(f)~~) \omega = \int_M (f\mathcal{L}_X + \mathcal{L}_X(f)) \omega~~ <~~/math~~>

	::<~~math~~> ~~= \int_M \mathcal{L}_X f\omega = \int_M \mathrm{d} \iota_X f\omega = \int_{\partial M} \iota_X f\omega</math>~~

	~~If ''f'' has [[compact support]]~~, ~~then the last integral vanishes~~, and ~~we have~~ the ~~desired result~~.

	~~==Laplace~~&~~ndash;de Rham operator==~~
	~~One may prove that the Laplace~~&~~ndash;de Rham operator is equivalent~~ to ~~the definition of the Laplace~~&~~ndash;Beltrami operator~~, ~~when acting~~ on ~~a scalar function ''f''~~. ~~This proof reads as:~~

	~~:<math>\Delta f =~~
	~~\mathrm{d}\delta f + \delta\,\mathrm{d}f =~~
	~~\delta\, \mathrm{d}f =~~
	~~\delta \, \partial_i f \, \mathrm{d}x^i</math>~~

	~~::<math> =~~
	~~- \mathrm{d}{\partial_i f \, \mathrm{d}x^i} =~~
	~~- *\mathrm{d}~~(~~\varepsilon_{i J}~~ ~~\sqrt{\|g\|}\partial^i f \, \mathrm{d}x^J)<~~/~~math>~~

	~~::<math> =~~
	~~- *\varepsilon_{i J} \, \partial_j~~
	~~(\sqrt{\|g\|}\partial^i f~~)\, ~~\mathrm{d} x^j \wedge \mathrm{d}x^J =~~
	~~- * \frac{1}{\sqrt{\|g\|}} \~~, ~~\partial_i (\sqrt{\|g\|}\,\partial^i f) \mathrm{vol}_n<~~/~~math>~~

	::<~~math~~> ~~= -\frac{1}{\sqrt{\|g\|}}\, \partial_i (\sqrt{\|g\|}\,\partial^i f),~~<~~/math~~>

	~~where vol_n; is the [[volume form]]~~ and ~~ε is~~ the ~~completely antisymmetric [[Levi-Civita symbol]]~~. ~~Note that~~ in the ~~above, the italic lower-case index ''i'~~' is a ~~single index~~, ~~whereas the upper-case Roman ''J'' stands for all of the remaining~~ (~~''n''-1~~) ~~indices~~. ~~Notice that the Laplace–de Rham operator is actually minus the Laplace–Beltrami operator; this minus sign follows from the conventional definition of the properties~~ of ~~the [[codifferential]]~~. ~~Unfortunately~~, ~~Δ is used to denote both; reader beware~~.

	== Properties ==
	~~Given scalar functions ''f''~~ and '~~'h''~~, and ~~a real number ''a''~~, the ~~Laplacian has the~~ property:

	~~:<math>\Delta(fh) = f \~~, ~~\Delta h + 2 \partial_i f \~~, ~~\partial^i h + h \, \Delta f~~.~~</math>~~

	~~===Proof===~~
	~~:<math>\Delta(fh) =~~
	~~\delta\~~,~~\mathrm{d}fh =~~
	~~\delta(f\~~,~~\mathrm{d}h + h\~~,~~\mathrm{d}f) =~~
	\mathrm{d}(f{\mathrm{d}h}) + \mathrm{d}(h{\mathrm{d}f})\;</math>

	~~:::<math> = (f\~~,~~\mathrm{d}\mathrm{d}h +~~
	~~\mathrm{d}f \wedge *\mathrm{d}h +~~
	~~\mathrm{d}h \wedge *\mathrm{d}f +~~
	h\,~~\mathrm{d}*\mathrm{d}f)~~
	~~</math>~~

	~~:::<math> =~~
	~~f\mathrm{d}\mathrm{d}h +~~
	(\mathrm{d}f \wedge \mathrm{d}h +
	~~\mathrm{d}h \wedge *\mathrm{d}f) +~~
	~~h\mathrm{d}\mathrm{d}f~~<~~/math~~>

	~~:::~~<~~math~~> ~~= f\~~, ~~\Delta h </math>~~

	~~::::<math> {} +~~
	*(\partial_i f \, ~~\mathrm{d}x^i \wedge~~
	~~\varepsilon_{jJ} \sqrt{\|g\|} \partial^j h \~~, ~~\mathrm{d}x^J +~~
	~~\partial_i h \~~, ~~\mathrm{d}x^i \wedge~~
	~~\varepsilon_{jJ} \sqrt{\|g\|} \partial^j f \, \mathrm{d}x^J~~) </~~math>~~

	~~::::<math> {} +~~
	~~h \, \Delta f<~~/~~math>~~

	~~:::<math> = f \, \Delta h +~~
	~~(\partial_i f \, \partial^i h +~~
	~~\partial_i h \, \partial^i f~~)~~{*\mathrm{vol}_n} +~~
	~~h \, \Delta f <~~/~~math>~~

	~~:::<math> = f \, \Delta h +~~
	~~2 \partial_i f \, \partial^i h +~~
	~~h \, \Delta f</math>~~

	~~where ''f'' and ''h'~~' are ~~scalar functions~~.

	~~{{DEFAULTSORT:Proofs involving~~ the ~~Laplace~~-~~Beltrami operator}}~~
	~~[[Category:Article proofs]]~~
	~~[[Category:Differential operators]]~~

en>Tobias Bergemann: /* Examples */ Fix typo.

2014-01-18T07:49:16Z

Examples: Fix typo.

New page

{{Unreferenced|date=December 2007}}

{{main|Laplace operator}}

==−div is adjoint to d==
The claim is made that −div is adjoint to ''d'':

:<math>\int_M df(X) \;\omega = - \int_M f \, \operatorname{div} X \;\omega </math>

Proof of the above statement:
:<math>\int_M (f\mathrm{div}(X) + X(f)) \omega = \int_M (f\mathcal{L}_X + \mathcal{L}_X(f)) \omega </math>

::<math> = \int_M \mathcal{L}_X f\omega = \int_M \mathrm{d} \iota_X f\omega = \int_{\partial M} \iota_X f\omega</math>

If ''f'' has [[compact support]], then the last integral vanishes, and we have the desired result.

==Laplace–de Rham operator==
One may prove that the Laplace–de Rham operator is equivalent to the definition of the Laplace–Beltrami operator, when acting on a scalar function ''f''. This proof reads as:

:<math>\Delta f =
\mathrm{d}\delta f + \delta\,\mathrm{d}f =
\delta\, \mathrm{d}f =
\delta \, \partial_i f \, \mathrm{d}x^i</math>

::<math> =
- *\mathrm{d}{*\partial_i f \, \mathrm{d}x^i} =
- *\mathrm{d}(\varepsilon_{i J} \sqrt{|g|}\partial^i f \, \mathrm{d}x^J)</math>

::<math> =
- *\varepsilon_{i J} \, \partial_j
(\sqrt{|g|}\partial^i f)\, \mathrm{d} x^j \wedge \mathrm{d}x^J =
- * \frac{1}{\sqrt{|g|}} \, \partial_i (\sqrt{|g|}\,\partial^i f) \mathrm{vol}_n</math>

::<math> = -\frac{1}{\sqrt{|g|}}\, \partial_i (\sqrt{|g|}\,\partial^i f),</math>

where vol_n; is the [[volume form]] and ε is the completely antisymmetric [[Levi-Civita symbol]]. Note that in the above, the italic lower-case index ''i'' is a single index, whereas the upper-case Roman ''J'' stands for all of the remaining (''n''-1) indices. Notice that the Laplace–de Rham operator is actually minus the Laplace–Beltrami operator; this minus sign follows from the conventional definition of the properties of the [[codifferential]]. Unfortunately, Δ is used to denote both; reader beware.

== Properties ==
Given scalar functions ''f'' and ''h'', and a real number ''a'', the Laplacian has the property:

:<math>\Delta(fh) = f \, \Delta h + 2 \partial_i f \, \partial^i h + h \, \Delta f.</math>

===Proof===
:<math>\Delta(fh) =
\delta\,\mathrm{d}fh =
\delta(f\,\mathrm{d}h + h\,\mathrm{d}f) =
*\mathrm{d}(f{*\mathrm{d}h}) + *\mathrm{d}(h{*\mathrm{d}f})\;</math>

:::<math> = *(f\,\mathrm{d}*\mathrm{d}h +
\mathrm{d}f \wedge *\mathrm{d}h +
\mathrm{d}h \wedge *\mathrm{d}f +
h\,\mathrm{d}*\mathrm{d}f)
</math>

:::<math> =
f*\mathrm{d}*\mathrm{d}h +
*(\mathrm{d}f \wedge *\mathrm{d}h +
\mathrm{d}h \wedge *\mathrm{d}f) +
h*\mathrm{d}*\mathrm{d}f</math>

:::<math> = f\, \Delta h </math>

::::<math> {} +
*(\partial_i f \, \mathrm{d}x^i \wedge
\varepsilon_{jJ} \sqrt{|g|} \partial^j h \, \mathrm{d}x^J +
\partial_i h \, \mathrm{d}x^i \wedge
\varepsilon_{jJ} \sqrt{|g|} \partial^j f \, \mathrm{d}x^J) </math>

::::<math> {} +
h \, \Delta f</math>

:::<math> = f \, \Delta h +
(\partial_i f \, \partial^i h +
\partial_i h \, \partial^i f){*\mathrm{vol}_n} +
h \, \Delta f </math>

:::<math> = f \, \Delta h +
2 \partial_i f \, \partial^i h +
h \, \Delta f</math>

where ''f'' and ''h'' are scalar functions.

{{DEFAULTSORT:Proofs involving the Laplace-Beltrami operator}}
[[Category:Article proofs]]
[[Category:Differential operators]]