Problem 30 - Advanced Engineering Mathematics

Question

In Problems \(25-32\), verify the given identity. Assume continuity of all partial derivatives. $$ \operatorname{div}(\text { curl } \mathbf{F})=0 $$

Step-by-Step Solution

Answer

The identity is verified: \( \nabla \cdot (\nabla \times \mathbf{F}) = 0 \).

Step 1: Understanding the Problem Statement

The exercise asks us to verify a mathematical identity involving vector calculus operations. Specifically, we need to show that the divergence of the curl of a vector field \( \mathbf{F} \) is equal to zero.

Step 2: Introduction to Vector Calculus Operations

We are dealing with two main operations: curl and divergence. The curl of a vector field \( \mathbf{F} = (F_1, F_2, F_3) \) is a vector operation that results in a vector, usually represented as \( abla \times \mathbf{F} \). The divergence of a vector field is a scalar operation, denoted as \( abla \cdot \mathbf{F} \).

Step 3: Write the Expression for Curl

The curl of \( \mathbf{F} = (F_1, F_2, F_3) \) is given by:\[abla \times \mathbf{F} = \begin{bmatrix}\hat{i} & \hat{j} & \hat{k} \\frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \F_1 & F_2 & F_3\end{bmatrix}= \left( \frac{\partial F_3}{\partial y} - \frac{\partial F_2}{\partial z}, \frac{\partial F_1}{\partial z} - \frac{\partial F_3}{\partial x}, \frac{\partial F_2}{\partial x} - \frac{\partial F_1}{\partial y} \right)\]

Step 4: Write the Divergence of the Curl

Now, we need to take the divergence of the curl of \( \mathbf{F} \). The curl \( \mathbf{C} \) is:\[ \mathbf{C} = \left( \frac{\partial F_3}{\partial y} - \frac{\partial F_2}{\partial z}, \frac{\partial F_1}{\partial z} - \frac{\partial F_3}{\partial x}, \frac{\partial F_2}{\partial x} - \frac{\partial F_1}{\partial y} \right) \]Thus, the divergence is:\[abla \cdot \mathbf{C} = \frac{\partial}{\partial x} \left( \frac{\partial F_3}{\partial y} - \frac{\partial F_2}{\partial z} \right) + \frac{\partial}{\partial y} \left( \frac{\partial F_1}{\partial z} - \frac{\partial F_3}{\partial x} \right) + \frac{\partial}{\partial z} \left( \frac{\partial F_2}{\partial x} - \frac{\partial F_1}{\partial y} \right)\]

Step 5: Simplify the Expression

Each term in the divergence expression represents derivatives of derivatives, specifically second derivatives of \( \mathbf{F} \). These can be rearranged and combined due to the symmetry and interchangeability of mixed partial derivatives (assuming continuity):- \( \frac{\partial^2 F_3}{\partial x \partial y} \) cancels with \( \frac{\partial^2 F_3}{\partial y \partial x} \)- \( \frac{\partial^2 F_2}{\partial y \partial z} \) cancels with \( \frac{\partial^2 F_2}{\partial z \partial y} \)- \( \frac{\partial^2 F_1}{\partial z \partial x} \) cancels with \( \frac{\partial^2 F_1}{\partial x \partial z} \)Thus, all terms cancel out, resulting in zero.

Step 6: Conclusion

After simplifying, we find that \( abla \cdot (abla \times \mathbf{F}) = 0 \). This confirms that the divergence of the curl of any vector field \( \mathbf{F} \) is indeed zero, verifying the identity.

Key Concepts

Divergence

In vector calculus, divergence is an important operation that measures a vector field's tendency to originate from or converge into a point. Simply put, it gives us a scalar value that represents whether a field is behaving like a source or sink at a given point.
The mathematical notation used for divergence is \( abla \cdot \mathbf{F} \), where \( \mathbf{F} \) is the vector field. This operation involves taking the dot product of the del operator, \( abla \), with the vector field.

• For a three-dimensional vector field \( \mathbf{F} = (F_1, F_2, F_3) \), the divergence is calculated as: \( abla \cdot \mathbf{F} = \frac{\partial F_1}{\partial x} + \frac{\partial F_2}{\partial y} + \frac{\partial F_3}{\partial z} \).
• Divergence is fundamental in many physical applications, such as fluid dynamics and electromagnetism, where it can denote fluid flow or electric field behavior.

Curl

The curl is another essential concept in vector calculus, which measures the rotation or the swirling pattern of a vector field. If you imagine a field containing arrows, the curl helps to understand how these arrows are rotating around a point.
The mathematical representation of curl is \( abla \times \mathbf{F} \). This vector operation results in another vector.

• For a three-dimensional vector field \( \mathbf{F} = (F_1, F_2, F_3) \), the curl is calculated using a determinant formula: \[ abla \times \mathbf{F} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \ F_1 & F_2 & F_3 \end{vmatrix} = \left( \frac{\partial F_3}{\partial y} - \frac{\partial F_2}{\partial z}, \frac{\partial F_1}{\partial z} - \frac{\partial F_3}{\partial x}, \frac{\partial F_2}{\partial x} - \frac{\partial F_1}{\partial y} \right) \]
• Physically, the curl quantifies the rotational effect or the tendency of a field to circulate around a point, commonly used in fluid mechanics and electromagnetism.

Partial Derivatives

Partial derivatives are derivatives used when dealing with functions of multiple variables. They allow us to measure how a function changes as only one of the variables is changed, keeping the others constant.
This concept is particularly useful in vector calculus when calculating divergence and curl, as these operations depend heavily on partial derivatives.

• For a function \( f(x, y, z) \), the partial derivative with respect to \( x \) is denoted as \( \frac{\partial f}{\partial x} \) and represents how \( f \) changes if only \( x \) changes.
• The same logic applies for \( \frac{\partial f}{\partial y} \) and \( \frac{\partial f}{\partial z} \).

When we assume continuity of partial derivatives, it means that mixed second-order partial derivatives can be interchangeably rearranged. This property is crucial for simplifying expressions like the divergence of the curl of a vector field, as it allows cancellation of terms. In essence, understanding partial derivatives and their properties is vital for mastering vector calculus concepts.