Question

For the general rotation field \(\mathbf{F}=\mathbf{a} \times \mathbf{r},\) where \(\mathbf{a}\) is a nonzero constant vector and \(\mathbf{r}=\langle x, y, z\rangle,\) show that \(\operatorname{curl} \mathbf{F}=2 \mathbf{a}\).

Short answer

Question: Prove that the curl of a vector field \(\mathbf{F}=\mathbf{a}\times\mathbf{r}\) is equal to \(2\mathbf{a}\). Answer: After calculating the curl of \(\mathbf{F}\), we find that \(\operatorname{curl} \mathbf{F} = 2\mathbf{a}\).

Step-by-step solution

Define the given vectors

We know that \(\mathbf{F}=\mathbf{a}\times\mathbf{r},\) and \(\mathbf{r}=\langle x,y,z \rangle\). Let \(\mathbf{a}=\langle a_1, a_2, a_3 \rangle\). Now, we can write \(\mathbf{F}\) in Cartesian coordinates, as follows: $$ \mathbf{F} = \mathbf{a} \times \mathbf{r} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ a_1 & a_2 & a_3 \\ x & y & z \end{vmatrix} $$

Calculate the cross product

Now, calculate the result of the cross product using the determinant definition: $$ \mathbf{F} = \langle (a_2z - a_3y), -(a_1z - a_3x), (a_1y - a_2x) \rangle $$

Find the curl of \(\mathbf{F}\)

The curl is defined as: $$ \operatorname{curl} \mathbf{F} = \nabla \times \mathbf{F} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ a_2z - a_3y & -(a_1z - a_3x) & a_1y - a_2x \end{vmatrix} $$

Calculate the curl

Now, calculate the determinant to find the curl: $$ \operatorname{curl} \mathbf{F} = \left\langle \frac{\partial (a_1y - a_2x)}{\partial z} - (-\frac{\partial (a_1z - a_3x)}{\partial y}), -\frac{\partial (a_1y - a_2x)}{\partial x} - \frac{\partial (a_2z - a_3y)}{\partial z}, -\frac{\partial (a_2z - a_3y)}{\partial y} - (-\frac{\partial (a_1z - a_3x)}{\partial x}) \right\rangle $$

Compute the partial derivatives

Now, compute the partial derivatives for each component of the curl: $$ \operatorname{curl} \mathbf{F} = \langle 2a_1, 2a_2, 2a_3 \rangle $$

Compare the curl with \(2\mathbf{a}\)

From step 5, we can see that \(\operatorname{curl} \mathbf{F} = 2\langle a_1, a_2, a_3 \rangle\). Since \(\mathbf{a}=\langle a_1, a_2, a_3 \rangle\), we can conclude that: $$ \operatorname{curl} \mathbf{F} = 2\mathbf{a} $$ So, it has been shown that \(\operatorname{curl} \mathbf{F} = 2 \mathbf{a}\).

Key concepts

Cross Product

Understanding the cross product is fundamental in vector calculus and physics. It's a binary operation on two vectors in three-dimensional space, resulting in a new vector that is perpendicular to the plane in which the original vectors lie. Imagine you have two vectors, let's call them vector A and vector B. When you calculate the cross product of A and B, denoted as A \times B, you are essentially finding a vector that points in the direction of your thumb if you were to hold your right hand with your fingers curling from A to B.

Mathematically, the cross product combines the components of the two vectors according to a specific rule, the determinant of a matrix. For vectors A = \ and B = \, the cross product is:\[ A \times B = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ a_1 & a_2 & a_3 \ b_1 & b_2 & b_3 \end{vmatrix} \]where \( \mathbf{i}, \mathbf{j}, \mathbf{k} \) are the unit vectors for the x, y, and z axes, respectively. The resulting vector's components are found by calculating the determinants of the 2x2 matrices formed by crossing out each row and column associated with the unit vector in question.

Partial Derivatives

Partial derivatives are at the heart of multivariable calculus. They measure how a function changes as one of its input variables changes, while keeping all other input variables constant. Imagine you're examining the effect of water temperature on the growth rate of an individual algae species in a multi-species pond. The partial derivative with respect to water temperature would tell you how much the growth rate of that algae varies with temperature, assuming light and nutrient levels stay the same.

In mathematical terms, if you have a function \( f(x, y, z) \) that depends on three variables, the partial derivative with respect to x is written as \( \frac{\partial}{\partial x} f(x, y, z) \) and is calculated by differentiating f with respect to x while treating y and z as constants. These operations are crucial for finding the rate of change in each direction, which in vector calculus leads us to gradients, curls, and divergences of vector fields.

Cartesian Coordinates

Cartesian coordinates provide a straightforward way to locate points in space by using perpendicular axes intersecting at a point called the origin. If you're navigating a city, think of Cartesian coordinates as street addresses. By knowing the x (east-west street) and y (north-south avenue) coordinates, you can pinpoint every location on the map, and if we add z (elevation), every point in three-dimensional space.

In the context of vector fields, every point \( \langle x, y, z \rangle \) represents a vector starting from the origin and reaching out to that point. This representation is essential because it allows us to specify functions and operations on vectors in a clear and manageable way, breaking them down into components related to each axis. In vector calculus, we often work with these coordinates to describe fields and perform operations like taking curls or divergences.

Vector Calculus

Vector calculus is a branch of mathematics that deals with vector fields and operations on vectors. It's like the calculus you learned in school but supercharged for dealing with multiple dimensions and directions all at once. Vector calculus encompasses a range of operations, including differentiation and integration of vector fields. In a real-world context, this could mean understanding how heat flows through a material in every direction, or how the pressure within a moving gas changes from point to point.

To express these complex, multi-directional changes, vector calculus uses specific tools, such as the gradient, curl, and divergence. The curl, for example, gives us a vector that describes the rotation or 'twist' of a vector field at a point. It's calculated using partial derivatives and has significance in physical applications like fluid flow and electromagnetism. These fundamental tools help mathematicians and scientists describe physical phenomena with precision and depth.