Problem 28 - Calculus Early Transcendentals

Question

Find \(\nabla \times(\nabla \times \mathbf{F})\) $$ \mathbf{F}(x, y, z)=y^{2} x \mathbf{i}-3 y z \mathbf{j}+x y \mathbf{k} $$

Step-by-Step Solution

Answer

\(\nabla \times (\nabla \times \mathbf{F}) = -2x\mathbf{i} + 2y\mathbf{j} - \mathbf{k}\)

Step 1: Find the Curl of F

First, we need to find the curl of the vector field \(\mathbf{F}(x, y, z) = y^2 x \mathbf{i} - 3 y z \mathbf{j} + xy \mathbf{k}\). The curl, denoted by \(abla \times \mathbf{F}\), is given by:\[abla \times \mathbf{F} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \ y^2 x & -3yz & xy \end{vmatrix}\]Calculating this determinant:- The \(\mathbf{i}\) component: \(\frac{\partial}{\partial y}(xy) - \frac{\partial}{\partial z}(-3yz) = x + 3y\).- The \(\mathbf{j}\) component: \(\frac{\partial}{\partial z}(y^2 x) - \frac{\partial}{\partial x}(xy) = 0 - y = -y\).- The \(\mathbf{k}\) component: \(\frac{\partial}{\partial x}(-3yz) - \frac{\partial}{\partial y}(y^2 x) = -3z - 2yx\).Thus, \(abla \times \mathbf{F} = (x + 3y)\mathbf{i} - y\mathbf{j} + (-3z - 2yx)\mathbf{k}\).

Step 2: Find the Curl of (Curl of F)

Now, we find \(abla \times (abla \times \mathbf{F})\). Denoting the curl found in Step 1 as \(\mathbf{G} = (x + 3y)\mathbf{i} - y\mathbf{j} - (3z + 2yx)\mathbf{k}\), we evaluate:\[abla \times \mathbf{G} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \ x + 3y & -y & -(3z + 2yx) \end{vmatrix}\]Calculating this determinant:- The \(\mathbf{i}\) component: \(\frac{\partial}{\partial y}(-3z - 2yx) - \frac{\partial}{\partial z}(-y) = -2x - 0 = -2x\).- The \(\mathbf{j}\) component: \(\frac{\partial}{\partial z}(x + 3y) - \frac{\partial}{\partial x}(-(3z + 2yx)) = 0 - (-2y) = 2y\).- The \(\mathbf{k}\) component: \(\frac{\partial}{\partial x}(-y) - \frac{\partial}{\partial y}(x + 3y) = 0 - 1 = -1\).Thus, \(abla \times (abla \times \mathbf{F}) = -2x\mathbf{i} + 2y\mathbf{j} - \mathbf{k}\).

Key Concepts

Curl of a Vector Field

The curl of a vector field is a fundamental concept in vector calculus that helps understand the rotational properties of a vector field. Essentially, it tells us how much a vector field rotates around a given point.

The curl is often represented by the symbol \(abla \times \mathbf{F}\), where \(abla\) is the del operator and \(\mathbf{F}\) is the vector field. The result is another vector field that describes the infinitesimal rotation at each point.

• For a three-dimensional vector field \(\mathbf{F}(x, y, z) = P\mathbf{i} + Q\mathbf{j} + R\mathbf{k}\), the curl is given by the determinant:

\[abla \times \mathbf{F} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \ P & Q & R \end{vmatrix}\] This expression defines the components of the curl as partial derivatives of the vector components \(P, Q, R\).

Understanding the curl is critical in physics, particularly in electromagnetism and fluid dynamics, where rotation and circulation of fields are significant. For example, in the exercise, we computed the curl of \(\mathbf{F}\) to find how it rotates in space.

Determinant Evaluation

Evaluating determinants is an essential skill when working with matrices and vectors in mathematics. It allows us to solve systems of equations, find areas and volumes, and, notably, compute the curl in vector calculus. Determinants offer a pattern for calculations that involve linear transformation properties.

For a 2x2 matrix, the determinant is simply \(ad - bc\) for a matrix \(\begin{bmatrix} a & b \ c & d \end{bmatrix}\). However, it becomes more intricate with a 3x3 matrix, such as when evaluating the curl.

The formula for a 3x3 determinant, used for curl in vector calculus, is:

• Choose cofactor expansion along a row or a column.
• Multiply by the respective minor (2x2 determinant).
• Consider alternating signs.

This approach gives the vector components for the curl:

• The \(\mathbf{i}\) component, determined by minor of the remaining matrix after removing the \(\mathbf{i}\)'s row and column.
• Same for \(\mathbf{j}\) and \(\mathbf{k}\) components.

Determinants abound in geometry and linear algebra, providing a robust toolkit for many computations in calculus and beyond.

Partial Derivatives

Partial derivatives are a way to extend the concept of derivatives to functions of several variables. Instead of measuring the rate of change of a function in one dimension, we assess how it changes in various directions in a multi-dimensional space.

Consider a function \(f(x, y, z)\). Its partial derivative with respect to \(x\), denoted \(\frac{\partial f}{\partial x}\), calculates the change of \(f\) as \(x\) varies, while keeping \(y\) and \(z\) constant.

Partial derivatives are crucial in many areas, including physics, engineering, and economics, because they reflect changes along one dimension while holding others fixed. They lay the groundwork for more complex concepts like gradients, divergences, and curls.

• In vector calculus, partial derivatives help calculate the components of the curl.
• Each component of the curl involves taking partial derivatives of vector field components.

The concept is fundamental because it extends calculus to functions of multiple variables, enabling analysis of more complex systems found in real-world applications.