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Q1: $$\vec{f}(x, y)=\left(y^{2}-2 x\right) i+2 x y j$$
Req: (1) $$\vec{F} $$ conservative vector field
(2) $$\vec{F}=\nabla f$$
Sol: (1) $$F_{1}(x, y)=y^{2}-2 x$$ $$F_{2}(x, y) = 2 x y$$
(2) $$\frac{d}{d y} F_{1}(x, y)=\frac{d}{d x} F_{2}(x, y)$$
(3) $$\frac{d}{d y} F_{1}(x, y)=2 y$$
$$\frac{d}{d x} F_{2}(x, y)=2 y$$
(4) $$\vec{F} $$: Conservative
(5) $$\vec{F}=\nabla f$$
$$f_{x}(x, y)=y^{2}-2 x$$ $$f_{y}(x, y)=2 x y$$
(6) integrate $$f_{x} \rightarrow(x)$$
$$f(x, y)=x y^{2}-x^{2}+g(y)$$
(7) $$f_{y}(x, y)=2 x y+g\prime(y)$$
$$g^{\prime}(y)=0$$
$$g(x)=C$$
$$f\left(x, y\right)=x y^{2}-x^{2}+g(y)$$
$$=x y^{2}-x^{2}+c$$
Q2: Given: $$\vec{f}\left(x, y\right)=(1+x y) e^{x y}\vec{ i}+x^{2} e^{x y} \vec{j} $$
C: $$\vec r(t)=\cos t \hat{i}+2 \sin t \vec{j} $$
$$0 \leq t \leq \frac{\pi}{2} $$
Req: $$F\left(x, y\right)=\nabla f$$
Sol: (1) $$f_{x} \vec i+f_{y} \vec j=(1+x y) e^{x y}\vec i+x^{2} e^{xy} \vec j$$
(2) $$f_{y}=x^{2} e^{x y} $$
$$f = \int x^{2} e^{x y} d y$$
$$f(x, y)=x e^{x y}+p(x) \rightarrow(1)$$
(3) $$f_x=(1+x y) e^{x y}+p^{\prime}(x)$$
(4) $$f_{x}=(1+x y) e^{x y} $$
$$p^{\prime}(x)=0$$
(5) $$f(x, y)=x e^{x y}+k$$
Q3: $$\int_{c} 2 x e^{-y} d x+\left(2 y-x^{2} e^{-y}\right) d y$$
$$C:(1,0) \rightarrow(2,1)$$
Req: (1) Line integral: independet
(2) Evaluat line integral
[1] (1) $$p=2 x e^{-y}=f_{x} $$
$$Q=2 y-x^{2} e^{-y}-f y$$
$$P_{y}=-2 x e^{-y}$$ $$Q_{x}=-2 x e^{-y}$$
(2) $$P_{y}=Q_{x} $$
conservative #
[2] (1) $$f(x, y)= \int 2 y-x^{2} e^{-y} d y$$
$$=y^{2}+x^{2} e^{-y}+h(x)$$
$$f_{x}=2 x e^{-y}=P$$
$$h(x) \rightarrow C$$
$$f(x, y)=y^{2}+x^{2} e^{-y}+C$$
(2) $$\int_{c} \vec {f} d r=f(2,1)-f(1,0)$$
$$=\left(1+\frac{4}{e}+c\right)-(1+c)=\frac{4}{e} $$
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Q1: $$\vec{f}(x, y)=\left(y^{2}-2 x\right) i+2 x y j$$
Req: (1) $$\vec{F} $$ conservative vector field
(2) $$\vec{F}=\nabla f$$
Sol: (1) $$F_{1}(x, y)=y^{2}-2 x$$
$$F_{2}(x, y) = 2 x y$$
(2) $$\frac{d}{d y} F_{1}(x, y)=\frac{d}{d x} F_{2}(x, y)$$
(3) $$\frac{d}{d y} F_{1}(x, y)=2 y$$
$$\frac{d}{d x} F_{2}(x, y)=2 y$$
(4) $$\vec{F} $$: Conservative
(5) $$\vec{F}=\nabla f$$
$$f_{x}(x, y)=y^{2}-2 x$$
$$f_{y}(x, y)=2 x y$$
(6) integrate $$f_{x} \rightarrow(x)$$
$$f(x, y)=x y^{2}-x^{2}+g(y)$$
(7) $$f_{y}(x, y)=2 x y+g\prime(y)$$
$$g^{\prime}(y)=0$$
$$g(x)=C$$
$$f\left(x, y\right)=x y^{2}-x^{2}+g(y)$$
$$=x y^{2}-x^{2}+c$$
Q2: Given: $$\vec{f}\left(x, y\right)=(1+x y) e^{x y}\vec{ i}+x^{2} e^{x y} \vec{j} $$
C: $$\vec r(t)=\cos t \hat{i}+2 \sin t \vec{j} $$
$$0 \leq t \leq \frac{\pi}{2} $$
Req: $$F\left(x, y\right)=\nabla f$$
Sol: (1) $$f_{x} \vec i+f_{y} \vec j=(1+x y) e^{x y}\vec i+x^{2} e^{xy} \vec j$$
(2) $$f_{y}=x^{2} e^{x y} $$
$$f = \int x^{2} e^{x y} d y$$
$$f(x, y)=x e^{x y}+p(x) \rightarrow(1)$$
(3) $$f_x=(1+x y) e^{x y}+p^{\prime}(x)$$
(4) $$f_{x}=(1+x y) e^{x y} $$
$$p^{\prime}(x)=0$$
(5) $$f(x, y)=x e^{x y}+k$$
Q3: $$\int_{c} 2 x e^{-y} d x+\left(2 y-x^{2} e^{-y}\right) d y$$
$$C:(1,0) \rightarrow(2,1)$$
Req: (1) Line integral: independet
(2) Evaluat line integral
[1] (1) $$p=2 x e^{-y}=f_{x} $$
$$Q=2 y-x^{2} e^{-y}-f y$$
$$P_{y}=-2 x e^{-y}$$
$$Q_{x}=-2 x e^{-y}$$
(2) $$P_{y}=Q_{x} $$
conservative #
[2] (1) $$f(x, y)= \int 2 y-x^{2} e^{-y} d y$$
$$=y^{2}+x^{2} e^{-y}+h(x)$$
$$f_{x}=2 x e^{-y}=P$$
$$h(x) \rightarrow C$$
$$f(x, y)=y^{2}+x^{2} e^{-y}+C$$
(2) $$\int_{c} \vec {f} d r=f(2,1)-f(1,0)$$
$$=\left(1+\frac{4}{e}+c\right)-(1+c)=\frac{4}{e} $$
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