Divergence and Curl

Divergence and Curl are differential operators in vector calculus. The divergence is a scalar operator applied to a 3D vector field, while the curl is a vector operator that measures the rotation of the field in three-dimensional space.

👁 Divergence-and-Curl

Divergence

Divergence is a vector calculus operator that measures the magnitude of a vector field's source or sink at a given point. In other words, it quantifies how much a vector field spreads out (diverges) or converges (compresses) at that point.

For a vector field F = (F₁, F₂, F₃) in three-dimensional space, where F₁, F₂​, and F₃ are the components of the vector field, the divergence of F is defined as:

div F = ∇.F = ∂/∂x(F₁) + ∂/∂y(F₂) + ∂/∂z(F₃)

where,

• ∇⋅ is Divergence Operator (Dot Product of Del Operator ∇ with Vector Field F)

Curl

Curl is a vector operator that describes the infinitesimal rotation of a vector field in three-dimensional space. In other words, it measures the tendency of the field to rotate around a point. The curl of a vector field provides information about the rotational motion or the "twisting" of the field lines around a given point.

For a vector field F = (F₁, F₂, F₃) in three-dimensional space, where F₁, F₂​, and F₃ are the components of the vector field, the curl of F is defined as:

curl F = ∇×F = (∂F₃/∂y - ∂F₂/∂z, ∂F₁/∂z - ∂F₃/∂x, ∂F₂/∂x - ∂F₁/∂y)

where,

• ∇× is Curl Operator (Cross Product of Del Operator ∇ with Vector Field F)

Divergence of Vector Field

The divergence of a vector field is a scalar field, denoted as "div." To calculate the divergence, you take the scalar product of the vector operator (∇) applied to the vector field, denoted as F(x, y). In two dimensions, for a vector field F(x, y), the divergence is given by:

In three dimensions, for a vector field F(x, y, z) represented as , the divergence is given by:

Divergence helps understand how a vector field's behavior changes concerning a point, providing valuable insights into the field's sources and sinks.

Curl of a Vector Field

The curl of a vector field is another vector field. To find the curl, we perform the vector product of the del operator applied to the vector field . Mathematically, it is represented as:

This can also be expressed as,

In simpler terms, the curl of a vector field indicates how the field rotates or circulates at each point in space.

Divergence of Curl

In a smooth vector field defined in a region of space (V), the divergence of the curl of is zero, i.e.

Proof of Divergence of Curl

Vector Field : Consider a vector field with components (F_x, F_y, F_z) defined in a region (V).

Curl of : Calculate the curl of using the cross product of the del operator and

: 

Use Cross-Product Identities

Apply Clairaut's Theorem

Since mixed partial derivatives are equal , the terms cancel each other.

The result simplifies to , confirming the divergence of the curl is zero. This theorem is a consequence of the vector calculus identities and plays a crucial role in understanding the relationships between different operations on vector fields.

Equations of Divergence and Curl

Curl Equation: The curl of a vector field is given by: 

Divergence Equation: The divergence of a vector field is calculated as: 

Divergence of Curl: The divergence of the curl of a vector field is always zero, i.e.

Curl of a Gradient: The curl of the gradient of a scalar function (f) is the zero vector, i.e.

These equations play a crucial role in vector calculus, describing the rotation and flow properties of vector fields, as well as the relationships between divergence and curl.

Related Articles

• Scalars and Vectors
• Vector Calculus
• Vector Algebra

Solved Examples on Divergence and Curl

Example 1: Consider the vector field . Find the divergence of and determine if the field is a source or a sink.

Solution:

Given,

• Vector Field 

For Divergence,

= 3y + 0 - 2x

So, the divergence of is ( 3y - 2x )

To determine if it's a source or sink, we need additional information about the region and boundary conditions.

• If in a region, it's a source
• If , it's a sink

Example 2: Given the vector field , calculate the curl of and interpret its meaning in terms of rotation and circulation.

Solution:

For Vector Field: ,

For Curl:

⇒ 

⇒ 

⇒ 

So, curl of is 

Practice Questions of Divergence and Curl

Q1. Given the vector field , calculate the divergence of and determine its nature (source, sink, or neither).

Q2. For the vector field , find the curl of and interpret its significance in terms of rotation.

Q3. Consider the vector field . Calculate both the divergence and curl of and assess any patterns or relationships between the two.

Q4. Given the vector field , compute the curl of and provide an interpretation of its physical significance.

Q5. For a vector field , prove that the divergence of the curl is zero.