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The given identities are verified by using operations of the del operator such as divergence and curl of the given vectors.
<h3>What are the divergence and curl of a vector field?</h3>The del operator is used for finding the divergence and the curl of a vector field.
The del operator is given by
Consider a vector field
Then the divergence of the vector F is,
div F = =
and the curl of the vector F is,
curl F = =
The given vector fields are:
and
1) Verifying the identity:
Consider L.H.S
⇒
⇒
⇒
⇒
Applying the dot product between these two vectors,
⇒ ...(1)
Consider R.H.S
⇒
So,
⇒
⇒
Then,
⇒ ...(2)
From (1) and (2),
2) Verifying the identity:
Consider L.H.S
⇒
⇒
⇒
Applying the cross product,
...(3)
Consider R.H.S,
⇒
So,
⇒
⇒
Then,
=
...(4)
Thus, from (3) and (4),
Learn more about divergence and curl of a vector field here:
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Disclaimer: The given question on the portal is incomplete.
Question: Let and
be differential vector fields and let a and b arbitrary real constants. Verify the following identities.