Question

Is there a vector field G on\({\mathbb{R}^3}\) such that curl \(G = \left\langle {xyz, - {y^2}z,y{z^2}} \right\rangle \)? Explain.

Short answer

The divergence is not zero hence such field does not exist.

Step-by-step solution

Theorem 11

Recall the statement of theorem 11 in the book. If the given field exists, then the divergence of its curl has to be zero.

The final solution using theorem 11

Using this fact, we calculate the divergence of the given field.

If the divergence is zero, then there may be such field and we might need to find it.

However :

\(\begin{array}{c}div((xyz, - {y^2}z,y{z^2})) = yz - 2yz + 2yz\\ = yz\end{array}\)

Since this is not zero, no such field exists.