Question

V=f(x,y,z); Prove that XVx=YVy=ZVz.​

Answer 1

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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