Divergence and Curl are mathematical operators, divergence is a differential operator, which is applied to the 3D vector-valued function. Whereas, the curl is a vector operator which defines the infinitesimal circulation of a vector field in the 3D Euclidean space. 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.
What is 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)
What is 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 (Fx, Fy, Fz) 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.
Read More,
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}~>~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.
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