V=f(x,y,z); Prove that XVx=YVy=ZVz.
Answers
Answer:
Step-by-step explanation:
Differential length vectors
dl = (dx,dy,dz) cart
= (dr,rdφ,dz)cyl
= (dρ,ρdθ,ρsin(φ)dφ)sph
Del Operator:
∇Φ = ˆ
i∂x + ˆ
j∂y + ˆ
k∂ ( z )Φ = rˆ∂r + ˆ
φr−1
∂φ + ˆ
k∂ ( z )Φ = ρˆ∂ ρ + (ρsin(θ))
−1 ˆ
φ∂φ + ˆ
θρ−1
∂ ( θ )Φ
∇ • V = ∂xVx + ∂y
Vy + ∂ ( z
Vz ) = r−1
∂r (rVr ) + r−1
∂φVφ + ∂ ( z
Vz )
= ρ−2
∂ ρ(ρ2
Vρ ) + (ρsin(θ))
−1
∂θ (sin(θ)Vθ ) + (ρsin(θ))
−1
∂ ( φVφ )
∇ ∧ V =
xˆ yˆ zˆ
∂x ∂y ∂z
Vx Vy Vz
= 1
r
rˆ rˆ
φ zˆ
∂r ∂φ ∂z
Vr rVφ Vz
= 1
ρ2 sin(θ)
ρˆ ρ ˆ
θ ρsin(θ)ˆ
φ
∂ ρ ∂θ ∂φ
Vρ ρVθ ρsin(θ)Vφ
∇ • ∇Φ = ∇2
Φ = ∂x
2 + ∂y
2 + ∂z
2 ( )Φ = r−1
∂r (r∂rΦ) + r−2
∂φ
2
Φ + ∂z
2 ( Φ)
= ρ−2
∂ ρ(ρ2
∂ ρΦ) + (ρ2 sin(θ))
−1
∂θ (sin(θ)∂θΦ) + (ρsin(θ))
−2
∂φ
2
( Φ)
∇ ∧ ∇ = 0
Green’s Theorem Q(x, y)dy + P(x,y)dx
enclo sin g
curve
∫ = (∂ xQ − ∂ yP)dxdy ∫∫
Divergence Theorem ∇ • V
Volume
∫∫∫ dτ = V• ndσ
enclo sin g
surface
∫∫
Stoke’s Theorem (∇ ∧ V)
surface
∫∫ • ndσ = V •d
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