Surface integral

In mathematics, a surface integral is a generalization of multiple integrals to integration over surfaces. It can be thought of as the double integral analog of the line integral. Given a surface, one may integrate over its scalar fields (that is, functions which return scalars as values), and vector fields (that is, functions which return vectors as values).

Surface integrals have applications in physics, particularly with the theories of classical electromagnetism.

Surface integrals of scalar fields

To find an explicit formula for the surface integral, we need to parameterize the surface of interest, S, by considering a system of curvilinear coordinates on S, like the latitude and longitude on a sphere. Let such a parameterization be x(s, t), where (s, t) varies in some region T in the plane. Then, the surface integral is given by

where the expression between bars on the right-hand side is the magnitude of the cross product of the partial derivatives of x(s, t), and is known as the surface element. The surface integral can also be expressed in the equivalent form

Podcasts:

PLAYLIST TIME:

Latest News for: surface integral

IAF to participate in Multi-national air exercise INIOCHOS-25 in Greece

Beijing News 30 Mar 2025
New Delhi [India], March 30(ANI) ... The exercise will integrate multiple air and surface assets from fifteen countries under realistic combat scenarios, designed to simulate modern-day air warfare challenges, the ministry's statement said on Sunday ... .

Controversial Marion County farmland dump pit at risk of failing, state says

Statesman Journal 30 Mar 2025
Here's what to know ... 26 ... 18, 2024 ... The bermed ‘pit’ is constructed and operated such that there is no surface discharge.”. “And your…speculation that the berm could ‘fail’ is belied by its continued integrity to this day,” Benke wrote.

Transform your garden this spring with these DIY tips

RTE 28 Mar 2025
For wooden furniture, sanding the surface creates a smooth base for paint to adhere to ... Sandtex Masonry Paint – A durable option for exterior walls and brick surfaces ... Clean the Surface – Remove dirt, grime, and any peeling paint.

PH, US to take 'bold steps' to improve Edca sites

Sun Star 28 Mar 2025
... (NMESIS) and unmanned surface vessels, integration into Exercise Balikatan, and special operations training in the Batanes Islands, which aims to support regional deterrence and defense modernization.

Revolutionary Bitcoin Core v29.0rc2 Release: Unleashing Enhanced Performance

Bitcoin World 28 Mar 2025
Let’s delve into some of the most noteworthy changes. Performance Optimizations ... These security enhancements are not always visible on the surface, but they are critical for maintaining the integrity and trustworthiness of the Bitcoin network ... ....

The Expanding Role of Red Teaming in Defending AI Systems

CRM Buyer 27 Mar 2025
Models trained on sensitive data, deployed in distributed environments, or integrated into decision-making processes create a broader attack surface ... Integrating AI red teaming into an ...

Blurring the line between rain and snow: Limits of meteorological classification

Science Daily 27 Mar 2025
Leveraging multi-source data integration rather than relying on surface weather data alone may offer improvements.

Pragmatic Semiconductor set to revolutionise NFC connectivity with sustainable flexible chips

Steel Technology 27 Mar 2025
Inlay manufacturers, label convertors and service bureaus can now empower brands to effortlessly integrate NFC functionality into products or packaging – even on curved surfaces – ...

India successfully test fires surface-to-air missile from Odisha coast

Hindustan Times 26 Mar 2025
Balasore Mar 26 India Wednesday successfully flight-test of indigenously-developed Vertically-Launched Short-Range Surface-to-Air Missile from the Integrated Test Range , Chandipur, off the coast of the Odisha, defence sources said.

India successfully flight-tests indigenously-developed vertically-launched Surface-to-Air Missile

The Siasat Daily 26 Mar 2025
... (DRDO) and the Indian Navy conducted the successful flight-test of indigenously-developed Vertically-Launched Short-Range Surface-to-Air Missile (VLSRSAM) on Wednesday, an official statement said.

The Benefits of Deck Sealing for Long-Term Protection Against Moisture and UV Damage

GetNews 25 Mar 2025
Without proper protection, wood surfaces can deteriorate, leading to costly repairs and reduced lifespan ... Over time, this can create hazardous conditions and reduce the overall integrity of the deck.

India must achieve complete self-reliance in cybersecurity

Hindustan Times 25 Mar 2025
Cybersecurity is no longer an auxiliary function but an integral component of national defence ... However, with the advent of Industry 4.0, IT-OT integration has expanded the attack surface ... Integrating ...

Motions to expel, censure Sen. Ellsworth fail

Raw Story 25 Mar 2025
Hertz called the floor debate a “historic moment” and laid out a timeline of events to illustrate a “consistent lack of ethics and integrity in the treatment of others and the public.”.

Norway’s arms supplies to Ukraine could have grave outcome

Blitz 25 Mar 2025
Earlier this year, Rustem Umerov (the Ukrainian Defense Minister) announced the possibility of integrating his country’s air defense systems with NASAMS. NASAMS, the Norwegian Advanced Surface-to-Air ...
×