Cálculo Iii

Derivadas Parciales
Tarea 1:
1.- Sea U=yFxy+ xGyx Calcular ∂U∂x , ∂U∂y
Fxy= f pero t=xy
Gyx= g pero r=yx

∂t∂x=1y
∂t∂y=-xy2
∂r∂x=-yx2
∂r∂y=1x

U=yf+xg∂U∂x=0f+y∂f∂x+1g+∂g∂x
=y∂f∂t∂t∂x+ g+x∂g∂r∂r∂x
=y∂f∂t1y+ g+x∂g∂r-yx2
=g+∂f∂t+-yx∂g∂r
∂U∂y=1f+y∂f∂y+0g+x∂g∂y
=f+y∂f∂t∂t∂y+0+x∂g∂r∂r∂y
=f+y∂f∂t-xy2+x∂g∂r1x
=f+∂g∂r + -xy∂f∂t

Tarea 2:
1.- SeaU=FxyGyx Calcular ∂2U∂x2

U=xy
v=yx

∂U∂x=1y
∂v∂x=-yx2

∂U∂x=∂f∂U∂U∂xβ+ γ∂g∂v∂v∂x
=∂f∂U1yβ+ γ∂g∂v-yx2

∂2U∂x=∂∂x∂f∂Uβy- γyx2∂g∂y
=∂f∂U∂∂xβy+βy ∂2f∂x∂U- ∂g∂v∂∂xγyx2+γyx2∂2g∂x∂v
=∂f∂U1y∂β∂x+βy ∂2f∂x∂U- ∂g∂vy∂∂xγx-2+γyx2 ∂2g∂x∂v
=∂f∂U1y∂g∂v∂v∂x+βy∂2f∂x∂U- ∂g∂vyγ(-2x-3+x-2∂γ∂x+ γyx2 ∂2g∂x∂v]
=∂f∂U1y∂g∂v-yx2+βy∂2f∂x∂U- ∂g∂vyγ(-2x-3+x-2∂f∂U∂U∂x+ γyx2∂2g∂x∂v]
=-∂f∂gx2∂u∂v+βy∂2f∂x∂U- yx3γ∂g∂v-2γx3+ ∂f∂U1yx-2+ γyx2 ∂2g∂x∂v
=1x2-∂f∂g∂u∂v- ∂fy∂U- γy ∂2g∂x∂v+ 1x3 -y∂g∂v-2γ+ βy∂2f∂x∂U

Tarea 3 de Operadores diferenciales.∇×F×G=G∙∇F-F∙∇G+∇∙GF-∇∙FG
F=f1,f2,f3
G=(g1,g2,g3)
F×G=f2g3-f3g2,f3g1-f1q3, f1g2-f2g1
∇×F×G=∂∂yf1g2-f2g1-∂∂zf3g1-f1q3, ∂∂zf2g3-f3g2-∂∂xf1g2-f2g1, ∂∂xf3g1-f1q3-∂∂yf2g3-f3g2∇×F×G=∂∂yf1g2-∂∂y(f2g1)-∂∂zf3g1+∂∂z(f1q3), ∂∂zf2g3-∂∂z(f3g2)-∂∂xf1g2+∂∂x(f2g1), ∂∂xf3g1-∂∂x(f1q3)-∂∂yf2g3+∂∂y(f3g2)
=f1∂g2∂y+∂g3∂z, f2∂g1∂x+∂g3∂z, f3∂g1∂x+∂g2∂y+g2∂f1∂y+g3∂f1∂z,g3∂f2∂z+g1∂f2∂x,g1∂f3∂x+g2∂f3∂y-g1∂f1∂y+g1∂f3∂z, g2∂f3∂z+g2∂f1∂x, g3∂f1∂x+g3∂f2∂y-f2∂g1∂y+f3∂g1∂z, f3∂g2∂z+f1∂g2∂x, f1∂g3∂x+f2∂g3∂y
=f1∂g1∂x+∂g2∂y+∂g3∂z, f2∂g1∂x+∂g2∂y+∂g3∂z,f3∂g1∂x+∂g2∂y+∂g3∂z+g1∂f1∂x+g2∂f1∂y+g3∂f1∂z,g2∂f2∂y+g3∂f2∂z+g1∂f2∂x, g1∂f3∂x+g2∂f3∂y+g3∂f3∂z-g1∂f1∂x+g1∂f2∂y+g1∂f3∂z, g2∂f3∂z+g2∂f2∂y+g2∂f1∂x, g3∂f1∂x+g3∂f2∂y+g3∂f3∂z-f1∂g1∂x+f2∂g1∂y+f3∂g1∂z, f3∂g2∂z+f2∂g2∂y+f1∂g2∂x, f1∂g3∂x+f2∂g3∂y+f3∂g3∂z=(f1,f2,f3∂g1∂x+∂g2∂y+∂g3∂z+g1∂∂x+g2∂∂y+g3∂∂zf1,f2,f3-(g1,g2,g3∂f1∂x+∂f2∂y+∂f3∂z-(f1∂∂x+f2∂∂y+f3∂∂z(g1,g2,g3)
Por lo tanto:...