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3.9: The Divergence Theorem
https://math.libretexts.org/Courses/Lake_Tahoe_Community_College/Interactive_Calculus_Q4/03%3A_Vector_Calculus/3.09%3A_The_Divergence_Theorem
We have examined several versions of the Fundamental Theorem of Calculus in higher dimensions that relate the integral around an oriented boundary of a domain to a “derivative” of that entity on the o...We have examined several versions of the Fundamental Theorem of Calculus in higher dimensions that relate the integral around an oriented boundary of a domain to a “derivative” of that entity on the oriented domain. In this section, we state the divergence theorem, which is the final theorem of this type that we will study.
16.4: Green’s Theorem
https://math.libretexts.org/Courses/Mission_College/Math_4A%3A_Multivariable_Calculus_v2_(Reed)/16%3A_Vector_Calculus/16.04%3A_Greens_Theorem
Green’s theorem is an extension of the Fundamental Theorem of Calculus to two dimensions. It has two forms: a circulation form and a flux form, both of which require region $D$ in the double integra...Green’s theorem is an extension of the Fundamental Theorem of Calculus to two dimensions. It has two forms: a circulation form and a flux form, both of which require region $D$ in the double integral to be simply connected. However, we will extend Green’s theorem to regions that are not simply connected. Green’s theorem relates a line integral around a simply closed plane curve $C$ and a double integral over the region enclosed by $C$ .
3.4E: Exercises
https://math.libretexts.org/Courses/De_Anza_College/Math_1D%3A_De_Anza/03%3A_Vector_Calculus/3.04%3A_Greens_Theorem/3.4E%3A_Exercises
These are homework exercises to accompany Chapter 16 of OpenStax's "Calculus" Textmap.
3.2: Line Integrals
https://math.libretexts.org/Courses/De_Anza_College/Math_1D%3A_De_Anza/03%3A_Vector_Calculus/3.02%3A_Line_Integrals
Line integrals have many applications to engineering and physics. They also allow us to make several useful generalizations of the Fundamental Theorem of Calculus. And, they are closely connected to t...Line integrals have many applications to engineering and physics. They also allow us to make several useful generalizations of the Fundamental Theorem of Calculus. And, they are closely connected to the properties of vector fields, as we shall see.
5.4: Green’s Theorem
https://math.libretexts.org/Courses/SUNY_Geneseo/Math_223_Calculus_3/05%3A_Vector_Calculus/5.04%3A_Greens_Theorem
Green’s theorem is an extension of the Fundamental Theorem of Calculus to two dimensions. It has two forms: a circulation form and a flux form, both of which require region $D$ in the double integra...Green’s theorem is an extension of the Fundamental Theorem of Calculus to two dimensions. It has two forms: a circulation form and a flux form, both of which require region $D$ in the double integral to be simply connected. However, we will extend Green’s theorem to regions that are not simply connected. Green’s theorem relates a line integral around a simply closed plane curve $C$ and a double integral over the region enclosed by $C$ .
5.9: The Divergence Theorem
https://math.libretexts.org/Courses/Mission_College/Math_4A%3A_Multivariable_Calculus_(Kravets)/05%3A_Vector_Calculus/5.09%3A_The_Divergence_Theorem
We have examined several versions of the Fundamental Theorem of Calculus in higher dimensions that relate the integral around an oriented boundary of a domain to a “derivative” of that entity on the o...We have examined several versions of the Fundamental Theorem of Calculus in higher dimensions that relate the integral around an oriented boundary of a domain to a “derivative” of that entity on the oriented domain. In this section, we state the divergence theorem, which is the final theorem of this type that we will study.
5.5: Green’s Theorem
https://math.libretexts.org/Courses/Mission_College/Math_4A%3A_Multivariable_Calculus_(Kravets)/05%3A_Vector_Calculus/5.05%3A_Greens_Theorem
Green’s theorem is an extension of the Fundamental Theorem of Calculus to two dimensions. It has two forms: a circulation form and a flux form, both of which require region $D$ in the double integra...Green’s theorem is an extension of the Fundamental Theorem of Calculus to two dimensions. It has two forms: a circulation form and a flux form, both of which require region $D$ in the double integral to be simply connected. However, we will extend Green’s theorem to regions that are not simply connected. Green’s theorem relates a line integral around a simply closed plane curve $C$ and a double integral over the region enclosed by $C$ .
5.5: Green’s Theorem
https://math.libretexts.org/Under_Construction/Purgatory/MAT-004A_-_Multivariable_Calculus_(Reed)/05%3A_Vector_Calculus/5.05%3A_Greens_Theorem
Green’s theorem is an extension of the Fundamental Theorem of Calculus to two dimensions. It has two forms: a circulation form and a flux form, both of which require region $D$ in the double integra...Green’s theorem is an extension of the Fundamental Theorem of Calculus to two dimensions. It has two forms: a circulation form and a flux form, both of which require region $D$ in the double integral to be simply connected. However, we will extend Green’s theorem to regions that are not simply connected. Green’s theorem relates a line integral around a simply closed plane curve $C$ and a double integral over the region enclosed by $C$ .
5.9: The Divergence Theorem
https://math.libretexts.org/Under_Construction/Purgatory/MAT-004A_-_Multivariable_Calculus_(Reed)/05%3A_Vector_Calculus/5.09%3A_The_Divergence_Theorem
We have examined several versions of the Fundamental Theorem of Calculus in higher dimensions that relate the integral around an oriented boundary of a domain to a “derivative” of that entity on the o...We have examined several versions of the Fundamental Theorem of Calculus in higher dimensions that relate the integral around an oriented boundary of a domain to a “derivative” of that entity on the oriented domain. In this section, we state the divergence theorem, which is the final theorem of this type that we will study.
16.5: Green’s Theorem
https://math.libretexts.org/Courses/City_College_of_San_Francisco/CCSF_Calculus/16%3A_Vector_Calculus/16.05%3A_Greens_Theorem
Green’s theorem is an extension of the Fundamental Theorem of Calculus to two dimensions. It has two forms: a circulation form and a flux form, both of which require region $D$ in the double integra...Green’s theorem is an extension of the Fundamental Theorem of Calculus to two dimensions. It has two forms: a circulation form and a flux form, both of which require region $D$ in the double integral to be simply connected. However, we will extend Green’s theorem to regions that are not simply connected. Green’s theorem relates a line integral around a simply closed plane curve $C$ and a double integral over the region enclosed by $C$ .
4.6: Vector Fields and Line Integrals: Work, Circulation, and Flux
https://math.libretexts.org/Bookshelves/Calculus/Supplemental_Modules_(Calculus)/Vector_Calculus/4%3A_Integration_in_Vector_Fields/4.6%3A_Vector_Fields_and_Line_Integrals%3A_Work%2C_Circulation%2C_and_Flux
This section demonstrates the practical application of the line integral in Work, Circulation, and Flux.

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