Divergence and Curl

Divergence and curl are fundamental concepts in vector calculus, extensively used to study the behavior of vector fields. These operations help describe how a vector field changes or flows through space, giving insights into the nature of the field itself. For a deeper understanding, let’s explore what divergence and curl mean, their mathematical formulations, and how to compute them.

1.0Divergence and Curl of a Vector Field

Divergence of a Vector Field  

Divergence measures how much a vector field spreads out or converges at a point. It is represented mathematically as:

$\mathrm{Divergence}(\nabla \cdot \mathbf{F})=\frac{\partial F_1}{\partial x}+\frac{\partial F_2}{\partial y}+\frac{\partial F_3}{\partial z}$

where $\mathbf{F}=F_1\hat{i}+F_2\hat{j}+F_3\hat{k}$ is a vector field in 3D space. The divergence helps determine whether a point acts as a source (positive divergence) or a sink (negative divergence).

Curl of a Vector Field

Curl, on the other hand, measures the rotational or swirling behavior of a vector field around a point. It is given by:

$\mathrm{Curl}(\nabla \times \mathbf{F})=\left|\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ \frac{\partial }{\partial x}& \frac{\partial }{\partial y}& \frac{\partial }{\partial z}\\ F_1& F_2& F_3\end{array}\right|$

$\mathrm{Curl}(\nabla \times \mathbf{F})=\left(\frac{\partial F_3}{\partial y}-\frac{\partial F_2}{\partial z}\right)\hat{i}+\left(\frac{\partial F_1}{\partial z}-\frac{\partial F_3}{\partial x}\right)\hat{j}+\left(\frac{\partial F_2}{\partial x}-\frac{\partial F_1}{\partial y}\right)\hat{k}$

It represents the axis and magnitude of rotation of the field around a given point.

2.0Physical Meaning of Gradient, Divergence, and Curl

Understanding the physical meaning of gradient, divergence, and curl is crucial for interpreting results beyond mathematical formulations:

• Gradient: Represents the rate and direction of change of a scalar field. For example, in thermodynamics, the temperature gradient points in the direction of the maximum increase of temperature.
• Divergence: Determines whether a vector field is a source or sink. In fluid dynamics, a positive divergence indicates fluid is emanating from a point, while a negative value indicates fluid converging.
• Curl: Describes the rotational nature or "twisting" of a field. In electromagnetism, the curl of an electric field can describe the magnetic field generated by changing electric fields.

3.0Divergence and Curl in Different Coordinate Systems

For practical applications, we often need to express these quantities in other coordinate systems like cylindrical and spherical coordinates:

1. Gradient, Divergence, and Curl in Cylindrical Coordinates:

Cylindrical Coordinates (r, \theta, z) are used when dealing with problems that exhibit symmetry about an axis.

Gradient: 

$\nabla f=\frac{\partial f}{\partial r}\hat{r}+\frac{1}{r}\frac{\partial f}{\partial \theta }\hat{\theta }+\frac{\partial f}{\partial z}\hat{z}$

Divergence:  

$\nabla \cdot \mathbf{F}=\frac{1}{r}\frac{\partial \left(r{F}_r\right)}{\partial r}+\frac{1}{r}\frac{\partial F_{\theta }}{\partial \theta }+\frac{\partial F_z}{\partial z}$

Curl:  

$\nabla \times \mathbf{F}=\left(\frac{1}{r}\frac{\partial F_z}{\partial \theta }-\frac{\partial F_{\theta }}{\partial z}\right)\hat{r}+\left(\frac{\partial F_r}{\partial z}-\frac{\partial F_z}{\partial r}\right)\hat{\theta }+\left(\frac{1}{r}\frac{\partial \left(r{F}_{\theta }\right)}{\partial r}-\frac{1}{r}\frac{\partial F_r}{\partial \theta }\right)\hat{z}$

Divergence and curl in cylindrical coordinates have different formulations due to the nature of the r and θ directions.

2. Gradient, Divergence, and Curl in Spherical Coordinates:

Spherical Coordinates $(r,\theta ,\varphi )$ are beneficial for problems with radial symmetry.

Gradient: 

$\nabla f=\frac{\partial f}{\partial r}\hat{r}+\frac{1}{r}\frac{\partial f}{\partial \theta }\hat{\theta }+\frac{1}{r\sin \theta }\frac{\partial f}{\partial \varphi }\hat{\varphi }$

Divergence:  

$\nabla \cdot \mathbf{F}=\frac{1}{r^2}\frac{\partial }{\partial r}\left(r^2{F}_r\right)+\frac{1}{r\sin \theta }\frac{\partial }{\partial \theta }\left(\sin \theta F_{\theta }\right)+\frac{1}{r\sin \theta }\frac{\partial F_{\varphi }}{\partial \varphi }$

Curl: 

$\nabla \times \mathbf{F}=\frac{1}{r\sin \theta }\left[\frac{\partial }{\partial \theta }\left(\sin \theta F_{\varphi }\right)-\frac{\partial F_{\theta }}{\partial \varphi }\right]\hat{r}+\frac{1}{r}\left[\frac{1}{\sin \theta }\frac{\partial F_r}{\partial \varphi }-\frac{\partial }{\partial r}\left(r{F}_{\varphi }\right)\right]\hat{\theta }+\frac{1}{r}\left[\frac{\partial }{\partial r}\left(r{F}_{\theta }\right)-\frac{\partial F_r}{\partial \theta }\right]\hat{\varphi }$

4.0Solved Examples of Divergence and Curl

Example 1: Find the divergence and curl of the vector field $\mathbf{F}=x^2\hat{i}+y^2\hat{j}+z^2\hat{k}$ .  

Solution: 

Divergence: 

$\nabla \cdot \mathbf{F}=\frac{\partial F_1}{\partial x}+\frac{\partial F_2}{\partial y}+\frac{\partial F_3}{\partial z}$

$\nabla \cdot \mathbf{F}=\frac{\partial \left(x^2\right)}{\partial x}+\frac{\partial \left(y^2\right)}{\partial y}+\frac{\partial \left(z^2\right)}{\partial z}$

$\nabla \cdot \mathbf{F}=2x+2y+2z$

Curl:

$\nabla \times \mathbf{F}=\left|\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ \frac{\partial }{\partial x}& \frac{\partial }{\partial y}& \frac{\partial }{\partial z}\\ x^2& y^2& z^2\end{array}\right|$

$\nabla \times \mathbf{F}=\left(\frac{\partial \left(z^2\right)}{\partial y}-\frac{\partial \left(y^2\right)}{\partial z}\right)\hat{i}+\left(\frac{\partial \left(x^2\right)}{\partial z}-\frac{\partial \left(z^2\right)}{\partial x}\right)\hat{j}+\left(\frac{\partial \left(y^2\right)}{\partial x}-\frac{\partial \left(x^2\right)}{\partial y}\right)\hat{k}$

$\nabla \times \mathbf{F}=0\hat{i}+0\hat{j}+0\hat{k}=0$

This indicates that the field has no rotation or swirling at any point in space.

Example 2: Find the divergence and curl of the vector field

$\mathbf{F}=(2x-y)\hat{i}+(x+3y)\hat{j}+(2z)\hat{k}$

Solution: 

Given Vector Field:

$\mathbf{F}=(2x-y)\hat{i}+(x+3y)\hat{j}+(2z)\hat{k}$

Here, the components of F are:

F₁ = 2x – y 

F₂ = x + 3y 

F₃ = 2z 

Divergence: 

The divergence of a vector field F is given by:

$\nabla \cdot \mathbf{F}=\frac{\partial F_1}{\partial x}+\frac{\partial F_2}{\partial y}+\frac{\partial F_3}{\partial z}$

Compute the partial derivatives:

$\frac{\partial F_1}{\partial x}=\frac{\partial (2x-y)}{\partial x}=2$

$\frac{\partial F_2}{\partial y}=\frac{\partial (x+3y)}{\partial y}=3$

$\frac{\partial F_3}{\partial z}=\frac{\partial (2z)}{\partial z}=2$

Thus, the divergence is:

$\nabla \cdot \mathbf{F}=2+3+2=7$

This indicates that the vector field has a net positive divergence, meaning it acts as a source in the given region.

Curl: 

The curl of a vector field F is defined as:

$\nabla \times \mathbf{F}=\left(\frac{\partial F_3}{\partial y}-\frac{\partial F_2}{\partial z}\right)\hat{i}+\left(\frac{\partial F_1}{\partial z}-\frac{\partial F_3}{\partial x}\right)\hat{j}+\left(\frac{\partial F_2}{\partial x}-\frac{\partial F_1}{\partial y}\right)\hat{k}$

Compute the partial derivatives:

$\frac{\partial F_3}{\partial y}=\frac{\partial (2z)}{\partial y}=0$

$\frac{\partial F_2}{\partial z}=\frac{\partial (x+3y)}{\partial z}=0$

$\frac{\partial F_1}{\partial z}=\frac{\partial (2x-y)}{\partial z}=0$

$\frac{\partial F_3}{\partial x}=\frac{\partial (2z)}{\partial x}=0$

$\frac{\partial F_2}{\partial x}=\frac{\partial (x+3y)}{\partial x}=1$

$\frac{\partial F_1}{\partial y}=\frac{\partial (2x-y)}{\partial y}=-1$

Plug these into the formula:

$\nabla \times \mathbf{F}=(0-0)\hat{i}+(0-0)\hat{j}+(1-(-1))\hat{k}=0\hat{i}+0\hat{j}+2\hat{k}$

Therefore, the curl is:

$\nabla \times \mathbf{F}=2\hat{k}$

This result indicates that the vector field has a constant rotational component along the z-axis.

Example 3: Find the divergence and curl of the vector field $\mathbf{F}=y\hat{i}+z\hat{j}+x\hat{k}$.

Solution: 

Divergence:  

$\nabla \cdot \mathbf{F}=\frac{\partial y}{\partial x}+\frac{\partial z}{\partial y}+\frac{\partial x}{\partial z}=0+0+0=0$

The divergence is zero, indicating no net flow into or out of any point in the field.

Curl:

$\nabla \times \mathbf{F}=\left(\frac{\partial x}{\partial y}-\frac{\partial z}{\partial z}\right)\hat{i}+\left(\frac{\partial y}{\partial z}-\frac{\partial x}{\partial x}\right)\hat{j}+\left(\frac{\partial z}{\partial x}-\frac{\partial y}{\partial y}\right)\hat{k}$

Simplifying, we get:

$\nabla \times {\mathbf{F}}_{=}(0-1)\hat{i}+(0-1)\hat{j}+(0-1)\hat{k}$

$\nabla \times \mathbf{F}=-(\hat{i}+\hat{j}+\hat{k})$

The curl is zero, indicating no rotational behavior in the field.

Example 4: Find the divergence and curl of the vector field $\mathbf{F}=r\hat{r}$ , where $r=\sqrt{x^2+y^2+z^2}$.

Solution: 

Divergence in Spherical Coordinates:

The divergence of a radial vector field $\mathbf{F}=f(r)\hat{r}$ in spherical coordinates is given by:

$\nabla \cdot \mathbf{F}=\frac{1}{r^2}\frac{\partial }{\partial r}\left(r^{2}f(r)\right)$

Substituting f(r) = r, we get:

$\nabla \cdot \mathbf{F}=\frac{1}{r^2}\frac{\partial }{\partial r}\left(r^3\right)=\frac{1}{r^2}\cdot 3r^2=3$

This indicates a uniform source throughout the field.

Curl:

Since F is purely radial, its curl is zero:

$\nabla \times \mathbf{F}=0$

This indicates no rotation or swirling behavior in the radial field.

Example 5: Find the divergence and curl of the vector field $\mathbf{F}=-y\hat{i}+x\hat{j}+0\hat{k}$ .

Solution: 

Divergence: 

$\nabla \cdot \mathbf{F}=\frac{\partial (-y)}{\partial x}+\frac{\partial x}{\partial y}+\frac{\partial 0}{\partial z}$

$\nabla \cdot \mathbf{F}=0+0+0=0$

This indicates no net source or sink.

Curl:

$\nabla \times \mathbf{F}=\left(\frac{\partial 0}{\partial y}-\frac{\partial x}{\partial z}\right)\hat{i}+\left(\frac{\partial (-y)}{\partial z}-\frac{\partial 0}{\partial x}\right)\hat{j}+\left(\frac{\partial x}{\partial x}-\frac{\partial (-y)}{\partial y}\right)\hat{k}$

Simplifying, we get:

$\nabla \times \mathbf{F}=0\hat{i}+0\hat{j}+2\hat{k}=2\hat{k}$

This shows that the field has a constant rotational component around the z-axis_.

5.0Practice Questions on Divergence and Curl 

1. Find the divergence and curl of the vector field:

$\mathbf{F}=x^2\hat{i}+y^2\hat{j}+z^2\hat{k}$

2. Find the divergence and curl of the vector field in cylindrical coordinates:

$\mathbf{F}=r^2\hat{r}+\theta \hat{\theta }+z\hat{z}$

3. Find the divergence and curl of the vector field:

$\mathbf{F}=\frac{x\hat{i}+y\hat{j}+z\hat{k}}{r^3},\text{ where }r=\sqrt{x^2+y^2+z^2}$

4. Evaluate the divergence and curl of the vector field:

$\mathbf{F}=-y\hat{i}+x\hat{j}+z\hat{k}$

Does this field have any sources, sinks, or rotational behavior?

5. For a given scalar function $\varphi =x^2+y^2+z^2$, find the gradient, divergence, and curl of the gradient of $\varphi$ .

6.0Sample Questions on Divergence and Curl

1. What is the Gradient, Divergence, and Curl in Cylindrical and Spherical Coordinates?

Ans: The formulas for calculating gradient, divergence, and curl differ when using cylindrical or spherical coordinates. This is due to the variation in the unit vectors and their dependence on the coordinates. For example, in cylindrical coordinates:

Gradient: 

$\nabla f=\frac{\partial f}{\partial r}\hat{r}+\frac{1}{r}\frac{\partial f}{\partial \theta }\hat{\theta }+\frac{\partial f}{\partial z}\hat{z}$

Divergence:  

$\nabla \cdot \mathbf{F}=\frac{1}{r}\frac{\partial \left(r{F}_r\right)}{\partial r}+\frac{1}{r}\frac{\partial F_{\theta }}{\partial \theta }+\frac{\partial F_z}{\partial z}$

Curl:  

$\nabla \times \mathbf{F}=\left(\frac{1}{r}\frac{\partial F_z}{\partial \theta }-\frac{\partial F_{\theta }}{\partial z}\right)\hat{r}+\left(\frac{\partial F_r}{\partial z}-\frac{\partial F_z}{\partial r}\right)\hat{\theta }+\left(\frac{1}{r}\frac{\partial \left(r{F}_{\theta }\right)}{\partial r}-\frac{1}{r}\frac{\partial F_r}{\partial \theta }\right)\hat{z}$

2. How are Divergence and Curl Calculated?

Ans: Divergence is calculated using the formula:

$\nabla \cdot \mathbf{F}=\frac{\partial F_1}{\partial x}+\frac{\partial F_2}{\partial y}+\frac{\partial F_3}{\partial z}$

where $\mathbf{F}=F_1\hat{i}+F_2\hat{j}+F_3\hat{k}$ is the given vector field.

Curl is calculated using the formula:

$\nabla \times \mathbf{F}=\left(\frac{\partial F_3}{\partial y}-\frac{\partial F_2}{\partial z}\right)\hat{i}+\left(\frac{\partial F_1}{\partial z}-\frac{\partial F_3}{\partial x}\right)\hat{j}+\left(\frac{\partial F_2}{\partial x}-\frac{\partial F_1}{\partial y}\right)\hat{k}$