Question

For \(\mathbf{r}=x \mathbf{i}+y \mathbf{j}+z \mathbf{k},\) evaluate $$\nabla \times(\mathbf{k} \times \mathbf{r})$$

Short answer

2\mathbf{k}

Step-by-step solution

- Understand the cross product

Recall that the cross product \(\mathbf{a} \times \mathbf{b}\) for vectors \(\mathbf{a}\) and \(\mathbf{b}\) in Cartesian coordinates can be evaluated using the determinant of a matrix. Specifically, \(\mathbf{k} \times \mathbf{r}\) can be found by considering the determinant of a 3x3 matrix with standard basis vectors, followed by the components of \(\mathbf{k}\) and \(\mathbf{r}\).

- Compute \ \mathbf{k} \times \mathbf{r} \

Given \(\mathbf{r} = x \mathbf{i} + y \mathbf{j} + z \mathbf{k}\), first compute the cross product with \(\mathbf{k} = 0 \mathbf{i} + 0 \mathbf{j} + 1 \mathbf{k}\):\[ \mathbf{k} \times \mathbf{r} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ 0 & 0 & 1 \ x & y & z \end{vmatrix} = -y \mathbf{i} + x \mathbf{j} \]

- Set up \ abla \times(\mathbf{k} \times \mathbf{r}) \

Next, set up and compute the curl of the resulting vector from the previous step, \(abla \times (-y \mathbf{i} + x \mathbf{j})\).

- Compute the curl

The curl of a vector \(\mathbf{A} = P \mathbf{i} + Q \mathbf{j} + R \mathbf{k}\) is computed as: \[abla \times \mathbf{A} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \ P & Q & R \end{vmatrix} \].\ In this problem, with \(P = -y, Q = x, R = 0\), evaluate the determinant: \[ abla \times (-y \mathbf{i} + x \mathbf{j}) = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \ -y & x & 0 \end{vmatrix}\]

- Evaluate the determinant

Evaluate the 3x3 determinant by expansion: \[ = \mathbf{i} \left( 0 - 0\right) - \mathbf{j} \left( 0 - 0 \right) + \mathbf{k} \left( \frac{\partial x}{\partial x} - \left( - \frac{\partial y}{\partial y} \right) \right) = 0 \mathbf{i} + 0 \mathbf{j} + (1 + 1)\mathbf{k} = 2 \mathbf{k}\]

Key concepts

cross product

The cross product is a unique operation in vector calculus. It involves two vectors, say \(\textbf{a}\) and \(\textbf{b}\). The result of the cross product is another vector that is perpendicular to both \(\textbf{a}\) and \(\textbf{b}\).

The cross product, \(\textbf{a} \times \textbf{b}\), can be computed using the determinant of a 3x3 matrix comprising standard basis vectors and the components of \(\textbf{a}\) and \(\textbf{b}\).

Let's see a concrete example. Consider vectors \(\textbf{a} = a_1\textbf{i} + a_2\textbf{j} + a_3\textbf{k}\) and \(\textbf{b} = b_1\textbf{i} + b_2\textbf{j} + b_3\textbf{k}\). The cross product can be formulated as:

\[ \textbf{a} \times \textbf{b} = \begin{vmatrix} \textbf{i} & \textbf{j} & \textbf{k} \ a_1 & a_2 & a_3 \ b_1 & b_2 & b_3 \end{vmatrix} \]

This determinant can be expanded to give:

\[ \textbf{a} \times \textbf{b} = \textbf{i}(a_2b_3 - a_3b_2) - \textbf{j}(a_1b_3 - a_3b_1) + \textbf{k}(a_1b_2 - a_2b_1) \]

In simple words, the cross product not only gives a new vector but also provides important geometric insights like the area of the parallelogram that the two input vectors span.

curl of a vector field

The curl of a vector field is a measure of the rotation or 'circulation' at any point in that field. If you imagine the vector field as a flow of fluid, the curl at a point represents how and how much the fluid is swirling around that point.

Mathematically, if a vector field is represented by \(\textbf{A} = P \textbf{i} + Q \textbf{j} + R \textbf{k}\), then the curl is defined as:

\[ abla \times \textbf{A} = \begin{vmatrix} \textbf{i} & \textbf{j} & \textbf{k} \ \frac{\rpartial}{\rpartial x} & \frac{\rpartial}{\rpartial y} & \frac{\rpartial{\rpartial z} \ P & Q & R \end{vmatrix} \]

The resulting vector field points in the direction of the axis of rotation and its magnitude tells us how much rotation or curl there is.

For our exercise, consider the field \(\textbf{A} = -y \textbf{i} + x \textbf{j}\). The curl is given by:

\[ abla \times (-y \textbf{i} + x \textbf{j}) = \begin{vmatrix} \textbf{i} & \textbf{j} & \textbf{k} \ \frac{\rpartial}{\rpartial x} & \frac{\rpartial}{\rpartial y} & \frac{\rpartial{\rpartial z} \ -y & x & 0 \end{vmatrix} \]

Expanding the determinant gives us the resulting vector field that describes the curl. As we simplify:

\[ abla \times (-y \textbf{i} + x \textbf{j}) = 2 \textbf{k} \]

This tells us the rotation is along the z-axis and has a magnitude of 2.

3x3 determinant

Understanding the 3x3 determinant is key to both cross product and curl. The determinant gives you a scalar value that encodes certain properties of the matrix.

A 3x3 matrix is set up as:

\[ A = \begin{pmatrix} a_{11} & a_{12} & a_{13} \ a_{21} & a_{22} & a_{23} \ a_{31} & a_{32} & a_{33} \end{pmatrix} \]

To find the determinant of this matrix, we use the formula:

\[ \text{det}(A) = a_{11}(a_{22}a_{33} - a_{23}a_{32}) - a_{12}(a_{21}a_{33} - a_{23}a_{31}) + a_{13}(a_{21}a_{32} - a_{22}a_{31}) \]

In our specific case, when evaluating the cross product or the curl of a vector field, the elements of our matrix are the components of our vectors and the partial derivatives.

For instance, let's revisit the curl operation once more:

\[ abla \times (-y \textbf{i} + x \textbf{j}) = \begin{vmatrix} \textbf{i} & \textbf{j} & \textbf{k} \ \frac{\rpartial}{\rpartial x} & \frac{\rpartial}{\rpartial y} & \frac{\rpartial}{\rpartial z} \ -y & x & 0 \end{vmatrix} \]

Expanding this determinant follows the same rule and gives us the vector field that describes the rotation.

The determinant’s usefulness does not stop here—it is foundational in understanding linear transformations, vector calculus operations like cross products, and in solving systems of linear equations.