The Relation between Greens theorem and Stokes theorem
![The Relation between Green’s theorem and Stokes’ theorem Closed Line integration and Surface (bounded The Relation between Green’s theorem and Stokes’ theorem Closed Line integration and Surface (bounded](https://slidetodoc.com/presentation_image_h/36fd3e05b4f1850829dd93c43261240d/image-1.jpg)
The Relation between Green’s theorem and Stokes’ theorem Closed Line integration and Surface (bounded by closed line) integration - The Extension and Special Case -
![The Similarity Green’s Theorem Stokes’ Theorem Both relate closed line integrals with surface integrals The Similarity Green’s Theorem Stokes’ Theorem Both relate closed line integrals with surface integrals](http://slidetodoc.com/presentation_image_h/36fd3e05b4f1850829dd93c43261240d/image-2.jpg)
The Similarity Green’s Theorem Stokes’ Theorem Both relate closed line integrals with surface integrals
![The Difference Green’s Theorem ST Stokes’ Theorem (1) cw GT Anti-cw ROTATION Stokes’ theorem The Difference Green’s Theorem ST Stokes’ Theorem (1) cw GT Anti-cw ROTATION Stokes’ theorem](http://slidetodoc.com/presentation_image_h/36fd3e05b4f1850829dd93c43261240d/image-3.jpg)
The Difference Green’s Theorem ST Stokes’ Theorem (1) cw GT Anti-cw ROTATION Stokes’ theorem is an extension of Green’s theorem (Stokes’ theorem is a higher dimension of Green’s theorem) Stokes’ theorem describe +ve and –ve surface orientation
![The Difference Green’s Theorem Stokes’ Theorem (2) ROTATION Green’s theorem is a special case The Difference Green’s Theorem Stokes’ Theorem (2) ROTATION Green’s theorem is a special case](http://slidetodoc.com/presentation_image_h/36fd3e05b4f1850829dd93c43261240d/image-4.jpg)
The Difference Green’s Theorem Stokes’ Theorem (2) ROTATION Green’s theorem is a special case of Stokes’ theorem (Stokes’ theorem for anti-clockwise closed line) Stokes’ theorem for +ve surface orientation = Green’s theorem GT Anti-cw
![Special case of Stokes’ theorem (Mathematical derivative from Stokes’ theorem to formularize Green’s theorem) Special case of Stokes’ theorem (Mathematical derivative from Stokes’ theorem to formularize Green’s theorem)](http://slidetodoc.com/presentation_image_h/36fd3e05b4f1850829dd93c43261240d/image-5.jpg)
Special case of Stokes’ theorem (Mathematical derivative from Stokes’ theorem to formularize Green’s theorem) z Unit normal vector for positive oriented surface: Let closed line curve C and surface S (bounded by closed line) in x-y plane Closed line in anti-clockwise direction. Choose any two point: 0 x Q P y Vector field refer to P and Q: Vector in x-y plane only
![Special case of Stokes’ theorem (Mathematical derivative from Stokes’ theorem to formularize Green’s theorem) Special case of Stokes’ theorem (Mathematical derivative from Stokes’ theorem to formularize Green’s theorem)](http://slidetodoc.com/presentation_image_h/36fd3e05b4f1850829dd93c43261240d/image-6.jpg)
Special case of Stokes’ theorem (Mathematical derivative from Stokes’ theorem to formularize Green’s theorem) From Stokes’ theorem formula: For closed line integration: …………………(1)
![Special case of Stokes’ theorem (Mathematical derivative from Stokes’ theorem to formularize Green’s theorem) Special case of Stokes’ theorem (Mathematical derivative from Stokes’ theorem to formularize Green’s theorem)](http://slidetodoc.com/presentation_image_h/36fd3e05b4f1850829dd93c43261240d/image-7.jpg)
Special case of Stokes’ theorem (Mathematical derivative from Stokes’ theorem to formularize Green’s theorem) From Stokes’ theorem formula: For surface integration: P and Q as a function of x and y =0 =0
![Special case of Stokes’ theorem (Mathematical derivative from Stokes’ theorem to formularize Green’s theorem) Special case of Stokes’ theorem (Mathematical derivative from Stokes’ theorem to formularize Green’s theorem)](http://slidetodoc.com/presentation_image_h/36fd3e05b4f1850829dd93c43261240d/image-8.jpg)
Special case of Stokes’ theorem (Mathematical derivative from Stokes’ theorem to formularize Green’s theorem) Then, …………………(2)
![Special case of Stokes’ theorem (Mathematical derivative from Stokes’ theorem to formularize Green’s theorem) Special case of Stokes’ theorem (Mathematical derivative from Stokes’ theorem to formularize Green’s theorem)](http://slidetodoc.com/presentation_image_h/36fd3e05b4f1850829dd93c43261240d/image-9.jpg)
Special case of Stokes’ theorem (Mathematical derivative from Stokes’ theorem to formularize Green’s theorem) Thus, Eq. (1) = Eq. (2) gives Green’s Theorem If we consider for clockwise direction of closed line, then we have surface in negative orientation, where , thus the relation gives:
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