P111 TD 1

f1(x,y) = xy
x2+y2f2(x,y) = x2y2
√yf3(x,y) = ylog(x) + xsin(y)
f1,f2f3
f1(x,y) = x2
yf2(x,y) = log|xy|f3(x,y) = xy=eylog|x|)f4(x,y) = (x2+y2)1
3
f5(x,y) = 1
√x2+y2f6(x,y,z) = zxlog|y|
w1= (x+ (y+ 1)2)dx + 2x(y+ 1)dy
w2= (y2−3)xdx + (1 + x2)ydy
w3=1
ydx −x y2dy
w4=y2xdx +x2ydy
w5=x
x2+y2dx +y x2+y2dy
w6=−4y2dx −x2dy
w1,w2,w3,w4,w5w6
w6
f1(x,y),f2(x,y),f3(x,y),f4(x,y)f5(x,y)
df1=w1,df2=w2,df3=w3,df4=w4, df5=w5

(x+y+1
y−y x2)dx + (x+y+1 x−x y2)dy
df =1−y2
(1+xy)2dx +1−x2 (1+xy)2dy
df
=P(x,y,z)dx +Q(x,y,z)dy +R(x,y,z)dz
P(x,y,z) = ϕ(x)(x2z+z2+ 2xz)Q(x,y,z) = 0
R(x,y,z) = ϕ(x)(x2+x)(x+ 2z)
ϕ(x)
ϕ(x)
F(P,V,T ) = P
V+P V T+P V T 2