Question 26.10: If Φ(x, y, z) is an arbitrary differentiable scalar field, s......
If Φ(x, y, z) is an arbitrary differentiable scalar field, show that curl (grad Φ) = ∇×(∇Φ) is always zero.
Given Φ = Φ (x, y, z) we have, by definition,
\nabla\phi={\frac{\partial\phi}{\partial x}}\mathbf{i}+{\frac{\partial\phi}{\partial y}}\mathbf{j}+{\frac{\partial\phi}{\partial z}}\mathbf{k}Then
curl(grad Φ) = ∇ × (∇Φ)
=\begin{vmatrix} i & j & k \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ \frac{\partial{\phi}}{\partial x} & \frac{\partial{\phi}}{\partial y} & \frac{\partial{\phi}}{\partial z} \end{vmatrix}
= (\frac{\partial}{\partial y}(\frac{\partial \phi }{\partial z}) – \frac{\partial}{\partial z} (\frac{\partial \phi}{\partial y}))i – (\frac{\partial}{\partial x}(\frac{\partial \phi}{\partial z}) – \frac{\partial}{\partial z} (\frac{\partial \phi}{\partial x}))j
+ (\frac{\partial}{\partial x} (\frac{\partial \phi}{\partial y}) – \frac{\partial}{\partial y}(\frac{\partial \phi}{\partial x}))k
Now, since \frac{\partial}{\partial x} (\frac{\partial \phi}{\partial y}) = \frac{\partial}{\partial y} (\frac{\partial \phi}{\partial x}) with similar results for the other mixed partial derivatives, it follows that
∇ ×(∇Φ) = 0
for any scalar field Φ whatsoever.