Answer to Question #229400 in Chemical Engineering for Lokika

Question

Show that curl(curl→{v})=grad(div→{v})-V^{2}→{v},→{v} is any vector and if isolenoidal, then find curl (curl →{v)?

Answer 1

The provided equation is :

 curl(curl→{v})=grad(div→{v})-V^{2}→{v},→{v}

And to show that

In that equation, any is a vector:

= ˆr r sin θ ∂ ∂θ (aφ sin θ) − ∂ ∂φ(aθ) + θˆ r sin θ ∂ ∂φ(ar ) − ∂ ∂r (aφr sin θ) + φˆ r ∂ ∂r (aθr ) − ∂ ∂θ (ar )

 And if isolenoidal,

To determine curl (curl →{v):

 

We can begin by summing the pair of the equation to obtain:

Then, together with the other pair as:

The curl equation for (curl →{v) therefore becomes: