Divergence and Curl

Last Updated : 26 Dec, 2023

Divergence and Curl are important concepts of Mathematics applied to vector fields. Divergence describes how a field behaves concerning or moving away from a point, while curl measures the rotational aspect of the field around a specific point. Divergence operators give scalar results whereas Curl operators give vector results.

In this article, we will learn about the divergence definition, curl definition, divergence of the vector field, curl of a vector field, and others in detail.

Divergence-and-Curl

Table of Content

What is Divergence?
What is Curl?
Divergence of Vector Field
Curl of a Vector Field
Divergence of Curl
Equations of Divergence and Curl

What is Divergence?

Divergence, in vector fields, tells us about the behavior of the field concerning or moving away from a specific point. Divergence of a vector field $\vec{\nabla}$ in either two-dimensional (R₂) or three-dimensional (R₃) space at a given point P gauges how much the field is “outflowing” at that point. When $\vec{P}$ represents the velocity of a fluid, its divergence at point P indicates the net rate of change, over time, of the fluid amount moving away from P (the tendency of the fluid to flow outward from P. Specifically, if the volume of fluid entering P equals the volume flowing out, the divergence at P is zero.

What is Curl?

In vector field analysis, the curl is our second focus, gauging the rotational aspect of the field around a specific point. Consider a fluid’s velocity represented by the vector field $\vec{F}$ . The curl of $\vec{F}$ at a point P is a vector indicating how particles near P tend to rotate around the axis aligned with this vector. The curl’s magnitude at P reflects the speed of rotation around this axis. Essentially, the curl signifies the vector field’s “spin” at that particular point. To visualize, think of placing a paddlewheel in the fluid at P, aligning its axis with the curl vector. The curl then characterizes the paddlewheel’s inclination to rotate.

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.

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Divergence and Curl Examples

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 $\nabla \cdot \vec{F} > 0$ in 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, the 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.

Divergence and Curl-FAQs

1. Define Divergence.

Divergence is a measure of how a vector field’s values spread out from a point. It signifies the rate of outward flux or expansion of a vector field at a specific location in space.

2. What is the Meaning of Curl?

Curl describes the rotational behavior of a vector field around a point. It indicates the tendency of the vector field to circulate or rotate about an axis at a given point.

3. What is Divergence of a Vector Field?

Divergence of a Vector Field $(\nabla \cdot \vec{F})$ : It measures how much the vector field is spreading out or converging at a particular point. Mathematically, it is the sum of the partial derivatives of the vector field’s components with respect to each coordinate.

4. What is Curl of a Vector Field?

Curl of a Vector Field $(\nabla \times \vec{F})$ is rotational aspect of the vector field around a specific point. The curl is computed by taking the cross product of the del operator with the vector field.

5. How to Find Divergence of a Vector Field?

To find the divergence $\nabla \cdot \vec{F}$ of a vector field $\vec{F}$ , calculate the sum of its partial derivatives with respect to each coordinate,

$\nabla \cdot \vec{F} = \frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y} + \frac{\partial F_z}{\partial z}$

7. Is the Divergence of Curl 0?

The divergence of the curl is always zero.

8. What is the Formula for div and curl?

The formulas for the divergence and curl of a vector field are

div F = ∂F₁/∂x + ∂F₂/∂y + ∂F₃/∂z
curl F = (∂F₃/∂y − ∂F₂/∂z, ∂F₁/∂z − ∂F₃/∂x, ∂F₂/∂x − ∂F₁/∂y)

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