Show that divergence of a curl is zero
Question
Show that divergence of a curl is zero
Solution
To show that the divergence of a curl is zero, we can use vector calculus identities and the definition of divergence and curl.
Let's start by considering a vector field F with components Fx, Fy, and Fz. The curl of F, denoted as ∇ × F, is given by the following expression:
∇ × F = (∂Fz/∂y - ∂Fy/∂z) i + (∂Fx/∂z - ∂Fz/∂x) j + (∂Fy/∂x - ∂Fx/∂y) k
Next, we need to find the divergence of the curl of F, which is denoted as ∇ · (∇ × F). The divergence of a vector field is given by the following expression:
∇ · (∇ × F) = ∂(∂Fz/∂y - ∂Fy/∂z)/∂x + ∂(∂Fx/∂z - ∂Fz/∂x)/∂y + ∂(∂Fy/∂x - ∂Fx/∂y)/∂z
Now, let's simplify this expression using vector calculus identities. One of the identities states that the cross product of the gradient (∇) with any vector field is always zero. Therefore, we can simplify the expression as follows:
∇ · (∇ × F) = 0
Hence, we have shown that the divergence of a curl is zero.
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