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Question-19341




Question Number 19341 by tawa tawa last updated on 09/Aug/17
Commented by tawa tawa last updated on 09/Aug/17
(1) Using Stoke′s theorem, find  ∮  y^2 dx + z^2 dy + x^2 dz ,  where  Γ is the closed  curve  A→B→C→A  and  A = (1, 0, 0),  B = (0, 0, 1),  C(0, 1, 0).
$$\left(\mathrm{1}\right)\:\mathrm{Using}\:\mathrm{Stoke}'\mathrm{s}\:\mathrm{theorem},\:\mathrm{find}\:\:\oint\:\:\mathrm{y}^{\mathrm{2}} \mathrm{dx}\:+\:\mathrm{z}^{\mathrm{2}} \mathrm{dy}\:+\:\mathrm{x}^{\mathrm{2}} \mathrm{dz}\:,\:\:\mathrm{where}\:\:\Gamma\:\mathrm{is}\:\mathrm{the}\:\mathrm{closed} \\ $$$$\mathrm{curve}\:\:\mathrm{A}\rightarrow\mathrm{B}\rightarrow\mathrm{C}\rightarrow\mathrm{A}\:\:\mathrm{and}\:\:\mathrm{A}\:=\:\left(\mathrm{1},\:\mathrm{0},\:\mathrm{0}\right),\:\:\mathrm{B}\:=\:\left(\mathrm{0},\:\mathrm{0},\:\mathrm{1}\right),\:\:\mathrm{C}\left(\mathrm{0},\:\mathrm{1},\:\mathrm{0}\right). \\ $$
Commented by tawa tawa last updated on 09/Aug/17
2)  Consider F^→  = (x + y − 4)i + (3xy)j + (2zz + z^2 )k,     s = (x, y, z) ∈ R^3 ∣z = 4 − x^2  − y^2  ,   z ≥ 0)  Find    ∫ ∫ ( ▽^→  ∙ F^→ ) ∙ d^→ s
$$\left.\mathrm{2}\right)\:\:\mathrm{Consider}\:\overset{\rightarrow} {\mathrm{F}}\:=\:\left(\mathrm{x}\:+\:\mathrm{y}\:−\:\mathrm{4}\right)\mathrm{i}\:+\:\left(\mathrm{3xy}\right)\mathrm{j}\:+\:\left(\mathrm{2zz}\:+\:\mathrm{z}^{\mathrm{2}} \right)\mathrm{k},\:\:\: \\ $$$$\left.\mathrm{s}\:=\:\left(\mathrm{x},\:\mathrm{y},\:\mathrm{z}\right)\:\in\:\mathbb{R}^{\mathrm{3}} \mid\mathrm{z}\:=\:\mathrm{4}\:−\:\mathrm{x}^{\mathrm{2}} \:−\:\mathrm{y}^{\mathrm{2}} \:,\:\:\:\mathrm{z}\:\geqslant\:\mathrm{0}\right) \\ $$$$\mathrm{Find}\:\:\:\:\int\:\int\:\left(\:\overset{\rightarrow} {\bigtriangledown}\:\centerdot\:\overset{\rightarrow} {\mathrm{F}}\right)\:\centerdot\:\overset{\rightarrow} {\mathrm{d}s} \\ $$

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