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• Notes

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) $\int_{c} \vec {f} d r=f(2,1)-f(1,0)$

$=\left(1+\frac{4}{e}+c\right)-(1+c)=\frac{4}{e}$