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16.4: Green's Theorem
https://math.libretexts.org/Courses/Misericordia_University/MTH_171-172%3A_Calculus_-_Early_Transcendentals_(Stewart)/16%3A_Vector_Calculus/16.04%3A_Green's_Theorem
\[\nonumber \begin{align} \oint_C Q(x, y)\,dy&=\int_{C_1}Q(x, y)\,dy+\int_{C_2}Q(x, y)\,dy \\ \nonumber &=\int_d^c Q(x_1(y), y)\,dy+\int_c^d Q(x_2(y), y)\,dy \\ \nonumber &=-\int_c^d Q(x_1(y), y)\,dy ...\[\nonumber \begin{align} \oint_C Q(x, y)\,dy&=\int_{C_1}Q(x, y)\,dy+\int_{C_2}Q(x, y)\,dy \\ \nonumber &=\int_d^c Q(x_1(y), y)\,dy+\int_c^d Q(x_2(y), y)\,dy \\ \nonumber &=-\int_c^d Q(x_1(y), y)\,dy + \int_c^d Q(x_2(y), y)\,dy \\ \nonumber &=\int_c^d (Q(x_2(y), y) - Q(x_1(y), y))\, dy \\ \nonumber &=\int_c^d \left ( Q(x, y) \Big |_{x=x_1(y)}^{x=x_2(y)} \right )\,dy \\ \nonumber &=\int_c^d \int_{x_1(y)}^{x_2(y)} \dfrac{∂Q(x, y)}{ ∂x}\,dx\,dy \text{ (by the Fundamental Theorem of Calculus)} \\ \…
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 $C$ .
3.7: Green's Theorem
https://math.libretexts.org/Bookshelves/Analysis/Complex_Variables_with_Applications_(Orloff)/03%3A_Multivariable_Calculus_(Review)/3.07%3A_Green's_Theorem
$C$ must be piecewise smooth (traversed so interior region $R$ is on the left) and piecewise smooth (a few corners are okay). Figure $\PageIndex{1}$ : Examples of piecewise smooth and piecewise s... $C$ must be piecewise smooth (traversed so interior region $R$ is on the left) and piecewise smooth (a few corners are okay). Figure $\PageIndex{1}$ : Examples of piecewise smooth and piecewise smooth regions. (CC BY-NC; Ümit Kaya) Theorem $\PageIndex{1}$ : Green's Theorem If the vector field $F = (M, N)$ is defined and differentiable on $R$ then $\oint_{C} F \cdot dr = \int \int_{R} \text{curl} F\ dA. \nonumber$ where the curl is defined as $\text{curl} F = (N_x - M_y)$
3.5: Divergence and Curl
https://math.libretexts.org/Courses/De_Anza_College/Math_1D%3A_De_Anza/03%3A_Vector_Calculus/3.05%3A_Divergence_and_Curl
Divergence and curl are two important operations on a vector field. They are important to the field of calculus for several reasons, including the use of curl and divergence to develop some higher-dim...Divergence and curl are two important operations on a vector field. They are important to the field of calculus for several reasons, including the use of curl and divergence to develop some higher-dimensional versions of the Fundamental Theorem of Calculus. In addition, curl and divergence appear in mathematical descriptions of fluid mechanics, electromagnetism, and elasticity theory, which are important concepts in physics and engineering.
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.
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.

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