Show that for any scaler field phi( x,y,z )
Vector del×(vector del phi ) = vector 0
To show that for any scalar field φ(x, y, z), the vector ∇ × (∇φ) = 0, we can use the properties of vector calculus. Here's the step-by-step process:
1. Start with the expression for the curl of a vector function F = (Fx, Fy, Fz), given by:
∇ × F = (∂Fz/∂y - ∂Fy/∂z) i + (∂Fx/∂z - ∂Fz/∂x) j + (∂Fy/∂x - ∂Fx/∂y) k
2. In our case, we have F = ∇φ = (∂φ/∂x, ∂φ/∂y, ∂φ/∂z). So, substituting the components, we get:
∇ × ∇φ = (∂(∂φ/∂z)/∂y - ∂(∂φ/∂y)/∂z) i + (∂(∂φ/∂x)/∂z - ∂(∂φ/∂z)/∂x) j + (∂(∂φ/∂y)/∂x - ∂(∂φ/∂x)/∂y) k
3. Now, let's evaluate each component of the curl:
(∂(∂φ/∂z)/∂y - ∂(∂φ/∂y)/∂z) = (∂²φ/∂y∂z - ∂²φ/∂z∂y) = 0 [using the symmetry of second partial derivatives]
(∂(∂φ/∂x)/∂z - ∂(∂φ/∂z)/∂x) = (∂²φ/∂z∂x - ∂²φ/∂x∂z) = 0 [same reason]
(∂(∂φ/∂y)/∂x - ∂(∂φ/∂x)/∂y) = (∂²φ/∂x∂y - ∂²φ/∂y∂x) = 0 [again, due to symmetry]
4. Combining all these components, we see that each component of the curl is equal to zero. Therefore, the vector ∇ × (∇φ) = 0.
Hence, we have shown that for any scalar field φ(x, y, z), the vector ∇ × (∇φ) is always equal to zero.