Problem 253 - Calculus Volume 3

Question

Find the curl of \(\mathbf{F}\) at the given point. \(\mathbf{F}(x, y, z)=x y z \mathbf{i}+y \mathbf{j}+z \mathbf{k}\) at (1,2,1)

Step-by-Step Solution

Answer

The curl of \( \mathbf{F} \) at (1, 2, 1) is \( 0 \mathbf{i} + 2 \mathbf{j} - 1 \mathbf{k} \).

Step 1: Understand Curl Formula

The curl of a vector field \( \mathbf{F}(x, y, z) = P \mathbf{i} + Q \mathbf{j} + R \mathbf{k} \) is given as \( abla \times \mathbf{F} = \left(\frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z}\right) \mathbf{i} + \left(\frac{\partial P}{\partial z} - \frac{\partial R}{\partial x}\right) \mathbf{j} + \left(\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\right) \mathbf{k} \).

Step 2: Identify Components P, Q, and R

From the vector field \( \mathbf{F}(x, y, z) = x y z \mathbf{i} + y \mathbf{j} + z \mathbf{k} \), we identify: \( P = x y z \), \( Q = y \), and \( R = z \).

Step 3: Compute Partial Derivatives

Calculate the necessary partial derivatives: - \( \frac{\partial R}{\partial y} = 0 \).- \( \frac{\partial Q}{\partial z} = 0 \).- \( \frac{\partial P}{\partial z} = x y \).- \( \frac{\partial R}{\partial x} = 0 \).- \( \frac{\partial Q}{\partial x} = 0 \).- \( \frac{\partial P}{\partial y} = xz \).

Step 4: Substitute in Curl Formula

Plug the partial derivatives into the curl formula to get: \( abla \times \mathbf{F} = \left(0 - 0\right) \mathbf{i} + \left(xy - 0\right) \mathbf{j} + \left(0 - xz\right) \mathbf{k} \).

Step 5: Simplify the Expression

Simplify the expression to: \( abla \times \mathbf{F} = 0 \mathbf{i} + xy \mathbf{j} - xz \mathbf{k} \).

Step 6: Evaluate at Given Point

Substitute the point \((1, 2, 1)\) into the simplified curl expression:\( abla \times \mathbf{F} = 0 \mathbf{i} + (1)(2) \mathbf{j} - (1)(1) \mathbf{k} = 0 \mathbf{i} + 2 \mathbf{j} - 1 \mathbf{k} \).

Key Concepts

Curl of a Vector Field

The concept of the curl in vector calculus is essential for understanding the behavior of vector fields. In simple terms, the curl measures the tendency of rotation at a given point in a vector field. Consider it as a way to determine how much "twist" a vector field has.
A vector field can be written generally as \( \mathbf{F}(x, y, z) = P \mathbf{i} + Q \mathbf{j} + R \mathbf{k} \), where \( P, Q, \) and \( R \) are the field components in the respective directions. The curl of this field is calculated using the formula:

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

This formula essentially involves taking partial derivatives of the vector components. The result is a new vector, which provides insight into the field's rotational nature at any point. Understanding how to apply this formula is crucial to finding the curl effectively.

Partial Derivatives

Partial derivatives are a core component of multivariable calculus, helping us understand how functions change with respect to one of several variables. In the context of the curl, they allow us to calculate how each vector component changes in response to small changes in one direction, while keeping other variables constant.
When finding the curl, you need to compute specific partial derivatives, such as \( \frac{\partial R}{\partial y} \), \( \frac{\partial Q}{\partial z} \), \( \frac{\partial P}{\partial z} \), among others. Each of these derivatives tells us how a component like \( R \) changes with a small change in \( y \), for example.

• Example: For our vector field \( \mathbf{F}(x, y, z) = x y z \mathbf{i} + y \mathbf{j} + z \mathbf{k} \), we calculated \( \frac{\partial R}{\partial y} = 0 \).

You should practice calculating these derivatives as they are essential for working with vector fields and finding the curl.

Vector Field Components

Breaking down a vector field into its components makes understanding and calculations more manageable. In vector calculus, fields are often represented as a sum of vector components along the \( \mathbf{i}, \mathbf{j}, \mathbf{k} \) axes which correspond to x, y, and z directions.For the vector field \( \mathbf{F}(x, y, z) = x y z \mathbf{i} + y \mathbf{j} + z \mathbf{k} \), we identify three core components:

• \( P = x y z \) as the x-component
• \( Q = y \) as the y-component
• \( R = z \) as the z-component

These components represent different parts of the vector field's behavior in different directions. Identifying them correctly is vital for applying operations like finding the curl, as it involves calculating derivatives of these components. By understanding each component's role, you can easily plug them into the needed formulas and grasp the field's behavior.