Problem 32 - Advanced Engineering Mathematics

Question

Verify the given identity. Assume continuity of all partial derivatives. \(\operatorname{curl}(\operatorname{curl} \mathbf{F}+\operatorname{grad} f)=\operatorname{curl}(\operatorname{curl} \mathbf{F})\)

Step-by-Step Solution

Answer

The identity is verified as the expression simplifies to the right-hand side.

Step 1: Understanding curl and grad

The "curl" of a vector field \( \mathbf{F} = \langle F_1, F_2, F_3 \rangle \) is given by \( abla \times \mathbf{F} = \left( \frac{\partial F_3}{\partial y} - \frac{\partial F_2}{\partial z}, \frac{\partial F_1}{\partial z} - \frac{\partial F_3}{\partial x}, \frac{\partial F_2}{\partial x} - \frac{\partial F_1}{\partial y} \right) \). The gradient "grad" of a scalar function \( f \) is \( abla f = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z} \right) \). The problem involves verifying an identity using these operations.

Step 2: Apply curl on the given expression

Write down the expression \( \operatorname{curl}(\operatorname{curl} \mathbf{F} + \operatorname{grad} f) \). Since curl is a linear operator, it distributes over addition: \( \operatorname{curl}(\operatorname{curl} \mathbf{F} + \operatorname{grad} f) = \operatorname{curl}(\operatorname{curl} \mathbf{F}) + \operatorname{curl}(\operatorname{grad} f) \).

Step 3: Compute curl of a gradient

Use the property of vector calculus that the curl of the gradient of any scalar function is always zero: \( \operatorname{curl}(\operatorname{grad} f) = \mathbf{0} \). This simplifies the expression to \( \operatorname{curl}(\operatorname{curl} \mathbf{F} + \operatorname{grad} f) = \operatorname{curl}(\operatorname{curl} \mathbf{F}) + \mathbf{0} = \operatorname{curl}(\operatorname{curl} \mathbf{F}) \).

Step 4: Confirm the identity

We have simplified the expression using the vector calculus identity to match the right side of the given identity: \( \operatorname{curl}(\operatorname{curl} \mathbf{F} + \operatorname{grad} f) = \operatorname{curl}(\operatorname{curl} \mathbf{F}) \). Thus, the identity is verified.

Key Concepts

Curl of a Vector Field

Understanding the curl of a vector field is crucial for grasping concepts in vector calculus related to rotation and circulation. When dealing with a vector field \( \mathbf{F} = \langle F_1, F_2, F_3 \rangle \), the curl is represented as a specific vector operation. It is notated as \( abla \times \mathbf{F} \) and performed using the cross product with the del operator. The resulting expression for the curl is

• \( \left( \frac{\partial F_3}{\partial y} - \frac{\partial F_2}{\partial z}, \frac{\partial F_1}{\partial z} - \frac{\partial F_3}{\partial x}, \frac{\partial F_2}{\partial x} - \frac{\partial F_1}{\partial y} \right) \).

The curl essentially measures the tendency of a vector field to circulate around a point. This concept is highly valuable in electromagnetism and fluid dynamics where it helps depict how a field "twists" or "rotates."
This rotational effect is often visualized as the amount of rotation a tiny paddlewheel placed in the field would experience. For example, in fluid dynamics, the curl can indicate how a fluid is curling or swirling.
Overall, understanding the vector field's curl allows us to assess important attributes of physical systems represented by the field.

Gradient

The gradient is a vector operation commonly applied to a scalar function in vector calculus. It converts the scalar function into a vector field, and its purpose is to show how the function changes in each spatial direction. If you have a scalar field represented by the function \( f(x, y, z) \), the gradient can be denoted by \( abla f \) and is given by the vector

• \( abla f = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z} \right) \).

This operation results in a vector field pointing in the direction of the greatest rate of increase of the function. The magnitude of this vector shows how steeply the function is increasing in that direction.
The gradient is often imagined as a slope or incline pointing upward, indicating where a value is most rapidly rising. In practical applications, finding the gradient can help determine where within a field the maximum rate of change occurs. For instance, in temperature fields, the gradient would tell you the direction and steepness of the temperature increase.
This tool is indispensable for tasks such as finding potentials and optimizing functions in economics, physics, and engineering.

Vector Identities

Vector identities are powerful tools in vector calculus. They allow mathematicians and engineers to simplify and manipulate complex vector equations. One key identity is that the curl of the gradient of any scalar field is zero:

• \( \operatorname{curl}(\operatorname{grad} f) = \mathbf{0} \).

This identity is fundamental because it implies that a scalar field's gradient never curls into a vector field. Essentially, gradients describe changes that lie along directional paths, whereas curl measures rotational motion.
There are various vector identities which combine operations such as divergence, gradient, and curl in different ways to achieve simplification or reveal characteristics of fields. For example, the curl of the curl identity, used in the original exercise, leverages the distribution property of the curl.

• \( \operatorname{curl}(\operatorname{curl} \mathbf{F} + \operatorname{grad} f) = \operatorname{curl}(\operatorname{curl} \mathbf{F}) \).

Understanding these identities can make complex calculations more manageable and help uncover underlying relationships within the field, providing insight into the structure and behavior of physical systems.

Partial Derivatives

Partial derivatives are a staple of multi-variable calculus and are necessary for evaluating how a function changes as each individual variable changes. For a function \( f(x, y, z) \), partial derivatives are represented as \( \frac{\partial f}{\partial x} \), \( \frac{\partial f}{\partial y} \), and \( \frac{\partial f}{\partial z} \). Each of these partial derivatives measures the rate of change of the function concerning one variable, assuming all other variables remain constant.
To better understand this, think of a hilly terrain where the elevation depends on the position, marked by coordinates. The partial derivative \( \frac{\partial f}{\partial x} \) would tell you how the elevation changes as you move purely in the x-direction, while not moving in y or z directions. Likewise, the other partial derivatives would inform you about the changes in their respective directions.
Partial derivatives play a significant role in forming the gradient, divergence, and curl, as they define the sensitivity of a function to its variables. They are also crucial in optimization problems, where maximizing or minimizing a function involves considering partial changes toward each variable independently. Understanding these derivatives provides the groundwork for analyzing and interpreting more complex two-dimensional and three-dimensional mathematical models.