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

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:

\text{div} \, \mathbf{F} = \frac{\partial F_1}{\partial x} + \frac{\partial F_2}{\partial y}

In three dimensions, for a vector field F(x, y, z) represented as F_1 \mathbf{i} + F_2 \mathbf{j} + F_3 \mathbf{k}, the divergence is given by:

\text{div} \, \mathbf{F} = \frac{\partial F_1}{\partial x} + \frac{\partial F_2}{\partial y} + \frac{\partial F_3}{\partial z}

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 \vec{F}(x, y, z). Mathematically, it is represented as:

\text{Curl } \vec{F}(x, y, z) = \nabla \times \vec{F}(x, y, z)

This can also be expressed as,

\text{Curl } \vec{F} = \left(\frac{\partial F_z}{\partial y} - \frac{\partial F_y}{\partial z}, \frac{\partial F_x}{\partial z} - \frac{\partial F_z}{\partial x}, \frac{\partial F_y}{\partial x} - \frac{\partial F_x}{\partial y}\right)

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 \vec{F}defined in a region of space (V), the divergence of the curl of \vec{F}is zero, i.e.

\nabla \cdot (\nabla \times \vec{F}) = 0

Proof of Divergence of Curl

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

Curl of \vec{F}: Calculate the curl of \vec{F}using the cross product of the del operator and

\vec{F}: \nabla \times \vec{F} = \left(\frac{\partial F_z}{\partial y} - \frac{\partial F_y}{\partial z}, \frac{\partial F_x}{\partial z} - \frac{\partial F_z}{\partial x}, \frac{\partial F_y}{\partial x} - \frac{\partial F_x}{\partial y}\right)

\nabla \cdot (\nabla \times \vec{F}) = \frac{\partial}{\partial x}\left(\frac{\partial F_y}{\partial x} - \frac{\partial F_x}{\partial y}\right) + \frac{\partial}{\partial y}\left(\frac{\partial F_z}{\partial y} - \frac{\partial F_y}{\partial z}\right) + \frac{\partial}{\partial z}\left(\frac{\partial F_x}{\partial z} - \frac{\partial F_z}{\partial x}\right)

Use Cross-Product Identities

\nabla \cdot (\nabla \times \vec{F}) = \frac{\partial^2 F_x}{\partial x^2} + \frac{\partial^2 F_y}{\partial y^2} + \frac{\partial^2 F_z}{\partial z^2} - \frac{\partial^2 F_x}{\partial y^2} - \frac{\partial^2 F_y}{\partial z^2} - \frac{\partial^2 F_z}{\partial x^2}

Apply Clairaut's Theorem

Since mixed partial derivatives are equal \frac{\partial^2 F_x}{\partial y^2} = \frac{\partial^2 F_y}{\partial x^2}\\ \frac{\partial^2 F_y}{\partial z^2} = \frac{\partial^2 F_z}{\partial y^2}\\ \frac{\partial^2 F_z}{\partial x^2} = \frac{\partial^2 F_x}{\partial z^2}, the terms cancel each other.

The result simplifies to \nabla \cdot (\nabla \times \vec{F})~=~0, 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 \vec{F}is given by: \nabla \times \vec{F} = (R_y - Q_z)\hat{i} + (P_z - R_x)\hat{j} + (Q_x - P_y)\hat{k}

Divergence Equation: The divergence of a vector field \vec{F}is calculated as: \nabla \cdot \vec{F} = P_x + Q_y + R_z

Divergence of Curl: The divergence of the curl of a vector field \vec{F}is always zero, i.e.

\nabla \cdot (\nabla \times \vec{F}) = 0

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

\nabla \times (\nabla f)~=~0

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 \vec{F} = 3xy\hat{i} + 2z\hat{j} - x^2\hat{k}. Find the divergence of \vec{F}and determine if the field is a source or a sink.

Solution:

Given,

• Vector Field \vec{F} = 3xy\hat{i} + 2z\hat{j} - x^2\hat{k}

For Divergence,

\nabla \cdot \vec{F} = \frac{\partial}{\partial x}(3xy) + \frac{\partial}{\partial y}(2z) + \frac{\partial}{\partial z}(-x^2)

= 3y + 0 - 2x

So, the divergence of \vec{F}is ( 3y - 2x )

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

• If \nabla \cdot \vec{F}~>~0in a region, it's a source
• If \nabla \cdot \vec{F}~<~0, it's a sink

Example 2: Given the vector field \vec{G}~=~(2y - z)\hat{i} + (x + z)\hat{j} + (y - x)\hat{k}, calculate the curl of \vec{G}and interpret its meaning in terms of rotation and circulation.

Solution:

For Vector Field: \vec{G}~=~(2y - z)\hat{i} + (x + z)\hat{j} + (y - x)\hat{k},

For Curl:

\nabla \times \vec{G}~=~\begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ 2y - z & x + z & y - x \end{vmatrix}

⇒ \left(\frac{\partial}{\partial y}(y - x) - \frac{\partial}{\partial z}(x + z)\right)\hat{i} - \left(\frac{\partial}{\partial x}(2y - z) - \frac{\partial}{\partial z}(y - x)\right)\hat{j} + \left(\frac{\partial}{\partial x}(x + z) - \frac{\partial}{\partial y}(2y - z)\right)\hat{k}

⇒ (1 + 1)\hat{i} - (-2 + 1)\hat{j} + (1 - 2)\hat{k}

⇒ 2\hat{i} + \hat{j} - \hat{k}

So, curl of \vec{G}is 2\hat{i} + \hat{j} - \hat{k}

Practice Questions of Divergence and Curl

Q1. Given the vector field \vec{A} = 4xy\hat{i} + 3z^2\hat{j} - 2xz\hat{k}, calculate the divergence of \vec{A}and determine its nature (source, sink, or neither).

Q2. For the vector field \vec{B} = (y + z)\hat{i} + (2x - z)\hat{j} + (x - y)\hat{k}, find the curl of \vec{B}and interpret its significance in terms of rotation.

Q3. Consider the vector field \vec{C} = x^2\hat{i} - y^2\hat{j} + xy\hat{k}. Calculate both the divergence and curl of \vec{C}and assess any patterns or relationships between the two.

Q4. Given the vector field \vec{D} = z\hat{i} + x\hat{j} + y\hat{k}, compute the curl of \vec{D}and provide an interpretation of its physical significance.

Q5. For a vector field \vec{E} = (y^2 - z^2)\hat{i} + (z^2 - x^2)\hat{j} + (x^2 - y^2)\hat{k}, prove that the divergence of the curl is zero.