Integraldisplay γ 1 y 2 dx x e y 2 dy

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Unformatted text preview: integraldisplay γ (1- y 2 ) dx + ( x + e y 2 ) dy = integraldisplay R (1 + 2 y ) dA change to = polars integraldisplay 2 π integraldisplay 1 (1 + 2 r sin θ ) r dr dθ = integraldisplay 2 π integraldisplay 1 ( r + 2 r 2 sin θ ) dr dθ = integraldisplay 2 π bracketleftbigg 1 2 r 2 + 2 3 r 3 sin θ bracketrightbigg 1 dθ = 1 2 integraldisplay 2 π dθ + a8 a8 a8 a8 a8 a8 a8 a8 a42 =0 2 3 integraldisplay 2 π sin θ dθ = 1 2 θ vextendsingle vextendsingle vextendsingle vextendsingle 2 π = π . 6. (a) (i) integraldisplay ∂D P Qdx + P Qdy Green prime s = Theorem integraldisplayintegraldisplay D parenleftbigg- ∂ ( PQ ) ∂y + ∂ ( PQ ) ∂x parenrightbigg dA = integraldisplayintegraldisplay D bracketleftbigg- parenleftbigg Q ∂P ∂y + P ∂Q ∂y parenrightbigg + parenleftbigg Q ∂P ∂x + P ∂Q ∂x parenrightbiggbracketrightbigg dA = integraldisplayintegraldisplay D bracketleftbigg Q parenleftbigg ∂P ∂x- ∂P ∂y parenrightbigg + P parenleftbigg ∂Q ∂x- ∂Q ∂y parenrightbiggbracketrightbigg dA . (ii) integraldisplay ∂D parenleftbigg Q ∂P ∂x- P ∂Q ∂x parenrightbigg dx + parenleftbigg P ∂Q ∂y- Q ∂P ∂y parenrightbigg dy Green prime s = Theorem integraldisplayintegraldisplay D ∂ parenleftBig P ∂Q ∂y- Q ∂P ∂y parenrightBig ∂x- ∂ ( Q ∂P ∂x- P ∂Q ∂x ) ∂y dA = integraldisplayintegraldisplay D parenleftbigg ∂P ∂x ∂Q ∂y + P ∂ 2 Q ∂x∂y- ∂Q ∂x ∂P ∂y- Q ∂ 2 P ∂x∂y- ∂Q ∂y ∂P ∂x- Q ∂ 2 P ∂y ∂x + ∂P ∂y ∂Q ∂x + P ∂ 2 Q ∂y ∂x parenrightbigg dA = integraldisplayintegraldisplay D parenleftbigg P parenleftbigg ∂ 2 Q ∂x∂y + ∂ 2 Q ∂y ∂x parenrightbigg- Q parenleftbigg ∂ 2 P ∂x, ∂y + ∂ 2 P ∂y ∂x parenrightbiggparenrightbigg dA = 2 integraldisplayintegraldisplay D parenleftbigg P ∂ 2 Q ∂x∂y- Q ∂ 2 P ∂x∂y parenrightbigg dA , since P, Q are assumed to be of class C 2 . (b) integraldisplay ∂D ∂f ∂y dx- ∂f ∂x dy Green prime s = Theorem integraldisplayintegraldisplay D bracketleftbigg ∂ ∂x parenleftbigg- ∂f ∂x parenrightbigg- ∂ ∂y parenleftbigg ∂f ∂y parenrightbiggbracketrightbigg dA =- integraldisplayintegraldisplay D parenleftbigg ∂ 2 f ∂x 2 + ∂ 2 f ∂y 2 parenrightbigg dA =- integraldisplayintegraldisplay D dA = 0....
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