Problem 40 - Calculus: Early Transcendental Functions

Question

If \(A\) is a constant vector and \(r=\langle x, y, z\rangle,\) prove that $$\nabla \times(\mathbf{A} \times \mathbf{r})=2 \mathbf{A}$$

Step-by-Step Solution

Answer

Therefore, the equation \(\nabla \times(\mathbf{A} \times \mathbf{r})=2 \mathbf{A}\) is proven to be true.

Step 1: Express the Cross Product A x r

First, we'll write \(\mathbf{A} \times \mathbf{r}\) using the determinant form. We know that the cross product of two vectors \(\mathbf{A} = A_x\mathbf{i}+A_y\mathbf{j}+A_z\mathbf{k} \) and \(\mathbf{r} = x\mathbf{i}+y\mathbf{j}+z\mathbf{k}\) is given by \( \mathbf{A} \times \mathbf{r} = (A_yz - A_z y) \mathbf{i} -(A_x z - A_z x)\mathbf{j} +(A_x y - A_y x) \mathbf{k}\).

Step 2: Compute the Curl

Next, we take the curl of the vector \(\nabla \times (\mathbf{A} \times \mathbf{r})\). Applying the operator curl to the vector obtained in step 1 we get:\(\nabla \times (\mathbf{A} \times \mathbf{r})= \nabla \times[(A_yz - A_z y) \mathbf{i} -(A_x z - A_z x)\mathbf{j} +(A_x y - A_y x) \mathbf{k}]\). This simplifies to:\((2A_y - 0 - 0)\mathbf{i} -(2A_x - 0 - 0)\mathbf{j} +(2A_z - 0 - 0) \mathbf{k}= 2A_x\mathbf{i}+2A_y\mathbf{j}+2A_z\mathbf{k}.\) Note that the divergence of a constant vector is zero hence the 0's in the expression.

Step 3: Simplify the Result

Finally, you can simplify the result from step 2 to get the final answer by factoring out 2 from the vector:\(2 (A_x\mathbf{i}+A_y\mathbf{j}+A_z\mathbf{k}) = 2\mathbf{A}\)

Key Concepts

Cross Product

Understanding the cross product in vector calculus is essential for analyzing the properties of vectors in three-dimensional space. The cross product, denoted by \times, is a binary operation on two vectors in three-dimensional space. It produces a third vector that is perpendicular to the plane containing the original vectors. To find the cross product of two vectors \textbf{A} and \textbf{B}, we can use the determinant of a matrix:

\[ \textbf{A} \times \textbf{B} = \begin{vmatrix} \textbf{i} & \textbf{j} & \textbf{k} \ A_x & A_y & A_z \ B_x & B_y & B_z \end{vmatrix} \]
Here, \textbf{i}, \textbf{j}, \textbf{k} are the unit vectors in the x, y, and z directions, respectively, and the subscripted letters represent the components of vectors \textbf{A} and \textbf{B}.

Properties of the Cross Product

• Orthogonality: The result is always orthogonal to the original vectors.
• Magnitude: The magnitude of the cross product is equal to the area of the parallelogram spanned by the original vectors.
• Direction: Given by the right-hand rule, which states that if you point your index finger along vector \textbf{A} and your middle finger along vector \textbf{B}, your thumb will point in the direction of \textbf{A} \times \textbf{B}.

In applying cross product to solve textbook exercises, it allows us to find the perpendicular vector to a plane described by two vectors and is often used in physics and engineering to determine torques and rotational forces.

Curl of a Vector Field

When dealing with vector fields in vector calculus, the concept of curl offers profound insight into the field's rotational tendencies. Curl is a vector operator that describes the infinitesimal rotation at a point within a vector field. Symbolically represented by \( abla \times \textbf{F} \), where \( abla \) is the del (or nabla) operator and \textbf{F} is the vector field, the curl can reveal the field's circulation density at that point.

To compute the curl, imagine taking the cross product of the gradient operator with the vector field. This operation can be visualized as the determinant of a matrix similar to how a cross product of two vectors is found:\[ abla \times \textbf{F} = \begin{vmatrix} \textbf{i} & \textbf{j} & \textbf{k} \ \frac{\textbf{partial}}{\textbf{partial} x} & \frac{\textbf{partial}}{\textbf{partial} y} & \frac{\textbf{partial}}{\textbf{partial} z} \ F_x & F_y & F_z \end{vmatrix} \]
In fluid dynamics, the curl of the velocity field can be used to describe the vorticity or the swirling patterns of fluid. Conversely, in electromagnetism, the curl of the magnetic field represents the induced electric field under a changing magnetic environment.

Application of Curl in Exercises

Exercises involving curl often require calculating the rotational effect of vector fields. The understanding of curl can, therefore, be directly applied to physical contexts such as determining the circulation of a field around a point, something that's crucial for problems in physics related to magnetism and fluid flow.

Gradient (Nabla) Operator

The gradient operator, famously known as the nabla operator and denoted by \( abla \), is a symbol used in vector calculus to denote a vector derivative operation. The term \( abla \textbf{F} \) signifies the gradient of a scalar field \textbf{F}, whereas \( abla \times \textbf{F} \) stands for the curl and \( abla \textbf{\textbf{cdot}} \textbf{F} \) represents the divergence when \textbf{F} is a vector field.

The gradient operator is composed of partial derivative operations with respect to each spatial dimension:\[ abla = \begin{pmatrix} \frac{\textbf{partial}}{\textbf{partial} x} \ \frac{\textbf{partial}}{\textbf{partial} y} \ \frac{\textbf{partial}}{\textbf{partial} z} \end{pmatrix} \]The gradient of a scalar field represents the rate and direction of change in the field's value most rapidly. In physical terms, it can be seen as the slope or incline of a scalar potential field.

Significance in Problem Solving

In terms of applying the gradient operator to solve exercises, the gradient can help determine the direction of the steepest ascent in a potential field. This is immensely useful in optimization problems, where locating the maximum or minimum values is essential. Additionally, gradients are vital in physics for describing forces, electric and magnetic fields, and more. Understanding how to use the nabla operator is thus critical for problems in virtually all fields of engineering and science.