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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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