Problem 18 - Advanced Engineering Mathematics

Question

Let \(\mathbf{a}\) be a constant voctor and \(\mathbf{r}=x \mathbf{i}+y \mathbf{j}+z \mathbf{k}\). Verify the given identity. \(\operatorname{curl} \mathbf{r}=\mathbf{0}\)

Step-by-Step Solution

Answer

The curl of the vector \(\mathbf{r}\) is \(\mathbf{0}\).

Step 1: Understand the Given Expression

The expression to verify is \(\operatorname{curl} \mathbf{r} = \mathbf{0}\), where \(\mathbf{r} = x \mathbf{i} + y \mathbf{j} + z \mathbf{k}\). This means we need to find the curl of the position vector \(\mathbf{r}\) and show that it equals the zero vector.

Step 2: Recall the Formula for Curl

The curl of a vector function \(\mathbf{F} = P \mathbf{i} + Q \mathbf{j} + R \mathbf{k}\) is given by \(abla \times \mathbf{F} = \left( \frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z} \right) \mathbf{i} + \left( \frac{\partial P}{\partial z} - \frac{\partial R}{\partial x} \right) \mathbf{j} + \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) \mathbf{k}\). Here, \(\mathbf{r}\) plays the role of \(\mathbf{F}\), so \(P=x\), \(Q=y\), and \(R=z\).

Step 3: Calculate the Curl Components

Calculate each component of the curl. - First component: \(\frac{\partial z}{\partial y} - \frac{\partial y}{\partial z} = 0 - 0 = 0\).- Second component: \(\frac{\partial x}{\partial z} - \frac{\partial z}{\partial x} = 0 - 0 = 0\).- Third component: \(\frac{\partial y}{\partial x} - \frac{\partial x}{\partial y} = 0 - 0 = 0\).

Step 4: Conclude the Calculation

Combine the zero components to express the curl of \(\mathbf{r}\) as \(abla \times \mathbf{r} = 0 \mathbf{i} + 0 \mathbf{j} + 0 \mathbf{k} = \mathbf{0}\). Thus, we have shown that \(\operatorname{curl} \mathbf{r} = \mathbf{0}\).

Key Concepts

Curl of a Vector

In vector calculus, the concept of the "curl" of a vector helps us understand how a vector field behaves in space, particularly its rotation. The curl is a vector that describes the infinitesimal rotation of a 3D vector field. It is typically used in physics, especially in electromagnetism and fluid mechanics, to describe rotational effects.
In mathematical terms, the curl of a vector field \( \mathbf{F} = P \mathbf{i} + Q \mathbf{j} + R \mathbf{k} \) is given by:

• Curl formula: \( abla \times \mathbf{F} = \left( \frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z} \right) \mathbf{i} + \left( \frac{\partial P}{\partial z} - \frac{\partial R}{\partial x} \right) \mathbf{j} + \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) \mathbf{k} \)

The curl tells us how much and in which direction the vector field tends to rotate as we move through the field. If the curl is zero everywhere, the vector field is described as "irrotational," meaning there is no net rotational effect at any point in the field. In our exercise, we computed the curl of the position vector \( \mathbf{r} = x \mathbf{i} + y \mathbf{j} + z \mathbf{k} \) and found it to be zero. This result signifies that the position vector has no rotation in space.

Position Vector

A position vector is a vector that originates from the origin of a coordinate system and points to a specific location in space. In a 3D Cartesian coordinate system, it is represented as \( \mathbf{r} = x \mathbf{i} + y \mathbf{j} + z \mathbf{k} \), where \( x, y, \) and \( z \) are the coordinates of the point.
The position vector is fundamental in various fields such as physics, where it is used to describe the position of an object in 3-dimensional space.

• Origin based: It's crucial that the position vector is defined relative to the origin, making it unique to the point it represents.
• Magical linking: The position vector provides a direct connection between any point in space and the origin, simplified as a linear path in a straight line.

In the context of our exercise, the position vector \( \mathbf{r} = x \mathbf{i} + y \mathbf{j} + z \mathbf{k} \) was examined to compute its curl, which resulted in zero, illustrating that the position vector itself does not contribute to any spatial rotation.

Partial Derivatives

Partial derivatives are essential in multivariable calculus, used to measure how a function changes as one of its variables is varied while others are held constant.
In our exercise, partial derivatives helped to determine the curl of the position vector. Here's how it works:

• Independent changes: They provide the rate of change of a function concerning one variable alone.
• Symbol representation: Denoted as \( \frac{\partial}{\partial x}, \) the partial derivative represents the change accordingly for each coordinate direction.

To calculate the curl of the position vector \( \mathbf{r} \), we took partial derivatives of its components with respect to various coordinates.
For instance, we evaluated expressions like \( \frac{\partial x}{\partial z} \) and \( \frac{\partial z}{\partial x} \) to examine rotational properties. In this case, each partial derivative resulted in zero, showing no change, leading to the conclusion that the curl is zero. Partial derivatives play an integral role in fields such as physics and engineering, where understanding how changes in variables affect complex systems is crucial.