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Magazine Chapter 9: Problem 23
Let \(\mathbf{a}\) be a constant voctor and \(\mathbf{r}=x \mathbf{i}+y \mathbf{j}+z \mathbf{k}\). Verify the given identity. $$ \nabla \times[(\mathbf{r} \cdot \mathbf{r}) \mathbf{a}]=2(\mathbf{r} \times \mathbf{a}) $$
Short Answer
Expert verified
The given identity is verified; both sides are equal, as shown in the steps.
Step by step solution
01
Understand the Problem Statement
We are given the identity \(abla \times[(\mathbf{r} \cdot \mathbf{r}) \mathbf{a}]=2(\mathbf{r} \times \mathbf{a})\). Here, \(\mathbf{r}\) is a position vector and \(\mathbf{a}\) is a constant vector. We need to verify this vector identity.
02
Express \(\mathbf{r} \cdot \mathbf{r}\) and \((\mathbf{r} \cdot \mathbf{r})\mathbf{a}\)
Calculate \(\mathbf{r} \cdot \mathbf{r}\):\[\mathbf{r} \cdot \mathbf{r} = x^2 + y^2 + z^2\]Now express \((\mathbf{r} \cdot \mathbf{r}) \mathbf{a}\) in terms of vector components assuming \(\mathbf{a} = a_i\mathbf{i} + a_j\mathbf{j} + a_k\mathbf{k}\):\[(x^2 + y^2 + z^2)(a_i \mathbf{i} + a_j \mathbf{j} + a_k \mathbf{k}) = a_i(x^2 + y^2 + z^2)\mathbf{i} + a_j(x^2 + y^2 + z^2)\mathbf{j} + a_k(x^2 + y^2 + z^2)\mathbf{k}\]
03
Compute \(\nabla \times [(\mathbf{r} \cdot \mathbf{r}) \mathbf{a}]\)
Using the expression obtained in Step 2, calculate the curl:- Express the derivatives: \(abla = \frac{\partial}{\partial x} \mathbf{i} + \frac{\partial}{\partial y} \mathbf{j} + \frac{\partial}{\partial z} \mathbf{k}\).- The curl formula: \(abla \times \mathbf{F} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \ F_x & F_y & F_z \end{vmatrix}\).- Substitute \(F = ((x^2 + y^2 + z^2)a_i, (x^2 + y^2 + z^2)a_j, (x^2 + y^2 + z^2)a_k)\) into the determinant and calculate.
04
Calculate \(\nabla \times \mathbf{F}\)
Compute the determinant to find \(abla \times \mathbf{F}\):- The \(x\)-component is \(\frac{\partial}{\partial y}((x^2 + y^2 + z^2)a_k) - \frac{\partial}{\partial z}((x^2 + y^2 + z^2)a_j) = 2(za_j - ya_k)\).- The \(y\)-component is \(\frac{\partial}{\partial z}((x^2 + y^2 + z^2)a_i) - \frac{\partial}{\partial x}((x^2 + y^2 + z^2)a_k) = 2(xa_k - za_i)\).- The \(z\)-component is \(\frac{\partial}{\partial x}((x^2 + y^2 + z^2)a_j) - \frac{\partial}{\partial y}((x^2 + y^2 + z^2)a_i) = 2(ya_i - xa_j)\).Thus, \(abla \times \mathbf{F} = 2((za_j - ya_k)\mathbf{i} + (xa_k - za_i)\mathbf{j} + (ya_i - xa_j)\mathbf{k})\).
05
Simplify \(2(\mathbf{r} \times \mathbf{a})\)
Calculate \(\mathbf{r} \times \mathbf{a}\) and multiply by 2:- \(\mathbf{r} \times \mathbf{a} = (x \mathbf{i} + y \mathbf{j} + z \mathbf{k}) \times (a_i \mathbf{i} + a_j \mathbf{j} + a_k \mathbf{k})\).- Using the cross product formula, find: - \( (ya_k - za_j)\mathbf{i}\) - \((za_i - xa_k)\mathbf{j}\) - \((xa_j - ya_i)\mathbf{k}\)- Multiply by 2 to match the expression found in Step 4: - \(2((ya_k - za_j)\mathbf{i} + (za_i - xa_k)\mathbf{j} + (xa_j - ya_i)\mathbf{k})\).
06
Verify that both sides are equal
Compare the results from Step 4 and Step 5. Notice that, in both cases, the final vector is:\[2 ((za_j - ya_k)\mathbf{i} + (xa_k - za_i)\mathbf{j} + (ya_i - xa_j)\mathbf{k})\]Since both calculations yield the same result, the given vector identity is verified.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
curl of a vector field
The concept of the curl of a vector field is crucial in vector calculus, particularly when dealing with fluid dynamics and electromagnetism.
It provides us with a measure of the rotational intensity at any point in a vector field.
Mathematically, the curl is expressed as the cross product of the del operator \(abla\) with a vector field \(\mathbf{F}\), denoted as \(abla \times \mathbf{F}\). This operator acts on the vector field and gives another vector field as a result.
It provides us with a measure of the rotational intensity at any point in a vector field.
Mathematically, the curl is expressed as the cross product of the del operator \(abla\) with a vector field \(\mathbf{F}\), denoted as \(abla \times \mathbf{F}\). This operator acts on the vector field and gives another vector field as a result.
- Components of the Curl: The curl in three dimensions is calculated using a determinant involving the unit vectors \(\mathbf{i}\), \(\mathbf{j}\), and \(\mathbf{k}\), the partial derivatives with respect to \(x\), \(y\), and \(z\), and the components \(F_x\), \(F_y\), and \(F_z\) of the vector field.
- Physical Interpretation: In physics, the curl essentially tells us how much and in what way a vector field is "circulating" around a point.
cross product
The cross product is an operation on two vectors in three-dimensional space, producing a third vector that is perpendicular to both.
This is fundamental in physics and engineering to determine a normal vector to two given vectors.
For vectors \(\mathbf{a} = a_i \mathbf{i} + a_j \mathbf{j} + a_k \mathbf{k}\) and \(\mathbf{b} = b_i \mathbf{i} + b_j \mathbf{j} + b_k \mathbf{k}\), the cross product \(\mathbf{a} \times \mathbf{b}\) is given by a determinant:
This is fundamental in physics and engineering to determine a normal vector to two given vectors.
For vectors \(\mathbf{a} = a_i \mathbf{i} + a_j \mathbf{j} + a_k \mathbf{k}\) and \(\mathbf{b} = b_i \mathbf{i} + b_j \mathbf{j} + b_k \mathbf{k}\), the cross product \(\mathbf{a} \times \mathbf{b}\) is given by a determinant:
- Calculate the Determinant: Set up the determinant with the first row as \(\mathbf{i}, \mathbf{j}, \mathbf{k}\), the second row with the components of \(\mathbf{a}\), and the third row with the components of \(\mathbf{b}\).
- Component Calculation: The determinant expands to deliver the components of the cross product vector. This includes the differences of the products of vector components across the rows and columns of the determinant.
- Perpendicular Vector: The resulting vector \(\mathbf{a} \times \mathbf{b}\) is perpendicular to both \(\mathbf{a}\) and \(\mathbf{b}\).
vector identity verification
Vector identity verification involves proving mathematical identities that relate different vector operations.
In our exercise, the identity \(abla \times[(\mathbf{r} \cdot \mathbf{r}) \mathbf{a}] = 2(\mathbf{r} \times \mathbf{a})\) needs to be confirmed. This is achieved through detailed algebraic manipulation and rigorous mathematical analysis.
In our exercise, the identity \(abla \times[(\mathbf{r} \cdot \mathbf{r}) \mathbf{a}] = 2(\mathbf{r} \times \mathbf{a})\) needs to be confirmed. This is achieved through detailed algebraic manipulation and rigorous mathematical analysis.
- Understanding Given Vectors: The problem begins by expressing the vectors and their dot and cross products, ensuring clarity on their components.
- Using Curl Properties: The verification process applies the properties of the curl and cross product to transform and compare both sides of the identity thoroughly.
- Matching Results: After performing the vector operations on both sides, obtaining identical expressions confirms the validity of the given identity.
