{"id":111,"date":"2021-07-30T17:14:54","date_gmt":"2021-07-30T17:14:54","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus3\/?post_type=chapter&p=111"},"modified":"2022-11-01T05:32:35","modified_gmt":"2022-11-01T05:32:35","slug":"introduction-to-stokes-theorem","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus3\/chapter\/introduction-to-stokes-theorem\/","title":{"raw":"Introduction to Stokes' Theorem","rendered":"Introduction to Stokes’ Theorem"},"content":{"raw":"

In this section, we study Stokes\u2019 theorem, a higher-dimensional generalization of Green\u2019s theorem. This theorem, like the Fundamental Theorem for Line Integrals and Green\u2019s theorem, is a generalization of the Fundamental Theorem of Calculus to higher dimensions. Stokes\u2019 theorem relates a vector surface integral over surface [latex]S[\/latex]\u00a0in space to a line integral around the boundary of\u00a0[latex]S[\/latex]. Therefore, just as the theorems before it, Stokes\u2019 theorem can be used to reduce an integral over a geometric object [latex]S[\/latex]\u00a0to an integral over the boundary of [latex]S[\/latex].<\/p>\r\n

In addition to allowing us to translate between line integrals and surface integrals, Stokes\u2019 theorem connects the concepts of curl and circulation. Furthermore, the theorem has applications in fluid mechanics and electromagnetism. We use Stokes\u2019 theorem to derive Faraday\u2019s law, an important result involving electric fields.<\/p>","rendered":"

In this section, we study Stokes\u2019 theorem, a higher-dimensional generalization of Green\u2019s theorem. This theorem, like the Fundamental Theorem for Line Integrals and Green\u2019s theorem, is a generalization of the Fundamental Theorem of Calculus to higher dimensions. Stokes\u2019 theorem relates a vector surface integral over surface [latex]S[\/latex]\u00a0in space to a line integral around the boundary of\u00a0[latex]S[\/latex]. Therefore, just as the theorems before it, Stokes\u2019 theorem can be used to reduce an integral over a geometric object [latex]S[\/latex]\u00a0to an integral over the boundary of [latex]S[\/latex].<\/p>\n

In addition to allowing us to translate between line integrals and surface integrals, Stokes\u2019 theorem connects the concepts of curl and circulation. Furthermore, the theorem has applications in fluid mechanics and electromagnetism. We use Stokes\u2019 theorem to derive Faraday\u2019s law, an important result involving electric fields.<\/p>\n\n\t\t\t

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