Problem 26 - Advanced Engineering Mathematics

Question

Verify the given identity. Assume continuity of all partial derivatives. $$ \nabla \times(\mathbf{F}+\mathbf{G})=\nabla \times \mathbf{F}+\nabla \times \mathbf{G} $$

Step-by-Step Solution

Answer

The identity \( \nabla \times (\mathbf{F} + \mathbf{G}) = \nabla \times \mathbf{F} + \nabla \times \mathbf{G} \) is verified by demonstrating it distributes over the addition.

Step 1: Understand the Expression

The expression \( abla \times (\mathbf{F} + \mathbf{G}) \) represents the curl of the sum of two vector fields \( \mathbf{F} \) and \( \mathbf{G} \). Our task is to verify that this is equal to the sum of the curls of the individual fields, i.e., \( abla \times \mathbf{F} + abla \times \mathbf{G} \).

Step 2: Recall the Curl Definition

The curl of a vector field \( \mathbf{A} = (A_1, A_2, A_3) \) is given by \( abla \times \mathbf{A} = \left( \frac{\partial A_3}{\partial x_2} - \frac{\partial A_2}{\partial x_3}, \frac{\partial A_1}{\partial x_3} - \frac{\partial A_3}{\partial x_1}, \frac{\partial A_2}{\partial x_1} - \frac{\partial A_1}{\partial x_2} \right) \).

Step 3: Apply the Curl to the Sum

Calculate \( abla \times (\mathbf{F} + \mathbf{G}) \):\( abla \times ((F_1 + G_1), (F_2 + G_2), (F_3 + G_3)) \).This results in:\[\left( \begin{array}{c}\frac{\partial (F_3 + G_3)}{\partial y} - \frac{\partial (F_2 + G_2)}{\partial z}, \\frac{\partial (F_1 + G_1)}{\partial z} - \frac{\partial (F_3 + G_3)}{\partial x}, \\frac{\partial (F_2 + G_2)}{\partial x} - \frac{\partial (F_1 + G_1)}{\partial y}\end{array}\right)\]

Step 4: Simplify Using Partial Derivative Properties

Use the linearity property of derivatives to distribute the derivatives:\[\begin{array}{c}\left(\frac{\partial F_3}{\partial y} + \frac{\partial G_3}{\partial y} - \frac{\partial F_2}{\partial z} - \frac{\partial G_2}{\partial z}, \\frac{\partial F_1}{\partial z} + \frac{\partial G_1}{\partial z} - \frac{\partial F_3}{\partial x} - \frac{\partial G_3}{\partial x}, \\frac{\partial F_2}{\partial x} + \frac{\partial G_2}{\partial x} - \frac{\partial F_1}{\partial y} - \frac{\partial G_1}{\partial y}\right)\end{array}\]

Step 5: Break Into Individual Curls

Recognize that this expression breaks into two separate curls:\((abla \times \mathbf{F}) + (abla \times \mathbf{G})\):1. \( abla \times \mathbf{F} = \left( \frac{\partial F_3}{\partial y} - \frac{\partial F_2}{\partial z}, \frac{\partial F_1}{\partial z} - \frac{\partial F_3}{\partial x}, \frac{\partial F_2}{\partial x} - \frac{\partial F_1}{\partial y} \right) \)2. \( abla \times \mathbf{G} = \left( \frac{\partial G_3}{\partial y} - \frac{\partial G_2}{\partial z}, \frac{\partial G_1}{\partial z} - \frac{\partial G_3}{\partial x}, \frac{\partial G_2}{\partial x} - \frac{\partial G_1}{\partial y} \right) \)

Step 6: Conclude the Verification

Since the result of Step 4 is a direct addition of the components calculated in Step 5, we verify the identity:\( abla \times (\mathbf{F} + \mathbf{G}) = abla \times \mathbf{F} + abla \times \mathbf{G} \) holds true.

Key Concepts

Partial Derivatives

Partial derivatives are an important concept in vector calculus, especially when dealing with functions of multiple variables. Imagine you have a function representing a surface, like the height of a mountain range. The height changes as you move along different directions (north, east, etc.), and a partial derivative measures the rate of change of the function along one specific direction while keeping other variables constant.

The process of finding a partial derivative involves differentiating the function with respect to one of the variables. For a function of two variables, such as \( f(x, y) \), partial derivatives would be \( \frac{\partial f}{\partial x} \) and \( \frac{\partial f}{\partial y} \). The notation \( \partial \) symbolizes a partial derivative as opposed to \( d \), which is used for ordinary derivatives.

Partial derivatives allow us to understand how a function behaves in each variable's direction. They are essential for gradient computations and understanding complex physical systems.

Curl of a Vector Field

The curl of a vector field is a crucial concept in vector calculus, often used in the context of fluid dynamics and electromagnetism. If you think about a vector field as representing the flow of water in a stream, the curl measures how much the flow "curls" or rotates at any given point.

Mathematically, the curl is a vector that describes the rotation. For a vector field \( \mathbf{A} = (A_1, A_2, A_3) \), the curl is calculated using the formula:

• \( abla \times \mathbf{A} = \left( \frac{\partial A_3}{\partial x_2} - \frac{\partial A_2}{\partial x_3}, \frac{\partial A_1}{\partial x_3} - \frac{\partial A_3}{\partial x_1}, \frac{\partial A_2}{\partial x_1} - \frac{\partial A_1}{\partial x_2} \right) \)

The result is a new vector field that gives the rotational activity at every point.

When you have multiple vector fields, as in the given exercise, taking the curl of their sum shows how the rotation distributes across both fields. It turns out that the curl of the sum is simply the sum of the curls, showcasing the linearity of the curl operation.

Differential Operators

Differential operators are mathematical tools that facilitate the manipulation and analysis of functions. In vector calculus, they play an essential role in identifying spatial variation and movement within a field.

Some common differential operators include the gradient (\( abla \)), divergence (\( abla \cdot \)), and curl (\( abla \times \)). Each has a specific function:

• The gradient assesses the direction and rate of fastest increase of a scalar field.
• The divergence measures a vector field's tendency to originate from or converge at a point, providing a scalar result.
• The curl produces a vector representing how much a field swirls or rotates around a point.

Understanding differential operators enables the dissection of vector fields into understandable components, helpful in physics and engineering contexts. They offer insight into the nature and behavior of fields, crucial for solving many practical problems.