Can a vector field have zero curl and zero divergence?

Curl and divergence are essentially “opposites” – essentially two “orthogonal” concepts. The entire field should be able to be broken into a curl component and a divergence component and if both are zero, the field must be zero.

Is the divergence of curl of a vector is zero?

Theorem 18.5. 1 ∇⋅(∇×F)=0. In words, this says that the divergence of the curl is zero. Recalling that gradients are conservative vector fields, this says that the curl of a conservative vector field is the zero vector.

How do you know if a vector field has zero divergence?

If the circle maintains its exact area as it flows through the fluid, then the divergence is zero. This would occur for both vector fields in (Figure). On the other hand, if the circle’s shape is distorted so that its area shrinks or expands, then the divergence is not zero.

What is a non zero curl?

With curl non-zero the arrows describing a vector field can form closed paths like the swirls in a fingerprint. Line integrals of vector fields along paths are path dependent when the field is not irrotational, that is when the curl is non-zero.

What is non zero divergence?

When the divergence of a field is zero at a particular point in space, it means that there is no source of the field at that point. A positive divergence means there is a source of the field and a negative divergence means that there’s a sink for the field.

Is curl of curl zero?

The curl of the gradient is the integral of the gradient round an infinitesimal loop which is the difference in value between the beginning of the path and the end of the path. In a scalar field there can be no difference, so the curl of the gradient is zero.

What does the zero value of the divergence of a vector field imply?

It means that if you take a very small volumetric space (assume a sphere for example) around a point where the divergence is zero, then the flux of the vector field into or out of that volume is zero. In other words, none of the arrows of the vector field will be piercing the sphere.

What is the divergence of curl of any vector field?

Divergence of Curl of any Vector Field is Zero. Or simply Curl of any Vector Field is Solenoidal. By this assumption if u take Curl of any Vector Field that is not irrotational, you will end up with another Vector Field which have Zero Divergence at every point.

What is the curl of a conservative vector field?

In words, this says that the divergence of the curl is zero. Recalling that gradients are conservative vector fields, this says that the curl of a conservative vector field is the zero vector. Under suitable conditions, it is also true that if the curl of F is 0 then F is conservative.

Can a non-zero constant field have zero curl and zero divergence?

One way your logic fails is that both curl and divergence are differentials of the field, and differentials don’t “see” constant terms. And consequently, the simplest counterexample to your claim is a non-zero constant field: It has zero curl and zero divergence everywhere, yet it is nowhere zero.

Can a field have a zero $x$ component and a 0$ divergence component?

The entire field should be able to be broken into a curl component and a divergence component and if both are zero, the field must be zero. I’m visualizing it like a vector in $\\mathbb{R}^2$. A vector cannot have a zero $x$ component and a zero $y$ component and still be non-zero.