To solve this problem, we need to use the properties of the divergence and the cross product.

The divergence of a cross product, ∇ · (c × r), can be simplified using the vector identity: ∇ · (A × B) = B · (∇ × A) - A · (∇ × B).

In this case, A corresponds to c (a constant vector) and B corresponds to r (xi + yj + zk).

Since c is a constant vector, its curl, ∇ × c, is zero. Therefore, the first term on the right side of the vector identity, B · (∇ × A), becomes r · (0) = 0.

The second term on the right side of the vector identity, - A · (∇ × B), becomes -c · (∇ × r). The curl of r, ∇ × r, is zero because r is a gradient of a scalar field. Therefore, this term also becomes -c · (0) = 0.

So, ∇ · (c × r) = 0 + 0 = 0.

Question

To solve this problem, we need to use the properties of the divergence and the cross product.

The divergence of a cross product, ∇ · (c × r), can be simplified using the vector identity: ∇ · (A × B) = B · (∇ × A) - A · (∇ × B).

In this case, A corresponds to c (a constant vector) and B corresponds to r (xi + yj + zk).

Since c is a constant vector, its curl, ∇ × c, is zero. Therefore, the first term on the right side of the vector identity, B · (∇ × A), becomes r · (0) = 0.

The second term on the right side of the vector identity, - A · (∇ × B), becomes -c · (∇ × r). The curl of r, ∇ × r, is zero because r is a gradient of a scalar field. Therefore, this term also becomes -c · (0) = 0.

So, ∇ · (c × r) = 0 + 0 = 0.

Knowee AI · Accepted Answer

To solve this problem, we need to use the properties of the divergence and the cross product.

The divergence of a cross product, ∇ · (c × r), can be simplified using the vector identity: ∇ · (A × B) = B · (∇ × A) - A · (∇ × B).

In this case, A corresponds to c (a constant vector) and B corresponds to r (xi + yj + zk).

Since c is a constant vector, its curl, ∇ × c, is zero. Therefore, the first term on the right side of the vector identity, B · (∇ × A), becomes r · (0) = 0.

The second term on the right side of the vector identity, - A · (∇ × B), becomes -c · (∇ × r). The curl of r, ∇ × r, is zero because r is a gradient of a scalar field. Therefore, this term also becomes -c · (0) = 0.

So, ∇ · (c × r) = 0 + 0 = 0.

Let r = xi + yj + zk and let c be a constant vector. Find ∇ · (c × r)

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