Question

The purpose in doing the following simple problems is to become familiar with the formulas we have discussed. So a good study method is to do them by hand and then check your results by computer. Compute the divergence and the curl of each of the following vector fields. $$\mathbf{r}=x \mathbf{i}+y \mathbf{j}$$

Short answer

Divergence: 2, Curl: 0.

Step-by-step solution

Identify the Vector Field

Given the vector field \(\textbf{r} = x \textbf{i} + y \textbf{j}\), identify the components. Here, \(F_1 = x\) and \(F_2 = y\).

Compute the Divergence

The divergence of a vector field \(\textbf{F} = F_1 \textbf{i} + F_2 \textbf{j}\) is given by \( abla \bullet \textbf{F} = \frac{\forall F_1}{\forall x} + \frac{\forall F_2}{\forall y} \). Calculate each partial derivative: \(\frac{\forall F_1}{\forall x} = \frac{\forall x}{\forall x} = 1\) and \(\frac{\forall F_2}{\forall y} = \frac{\forall y}{\forall y} = 1\). Hence, \( abla \bullet \textbf{r} = 1 + 1 = 2 \).

Compute the Curl

In 2D, the curl of a vector field \( \textbf{F} = F_1 \textbf{i} + F_2 \textbf{j} \) is given by \( abla \times \textbf{F} = \frac{\forall F_2}{\forall x} - \frac{\forall F_1}{\forall y} \). Compute each partial derivative: \(\frac{\forall F_2}{\forall x} = \frac{\forall y}{\forall x} = 0\) and \(\frac{\forall F_1}{\forall y} = \frac{\forall x}{\forall y} = 0\). Therefore, \( abla \times \textbf{r} = 0 - 0 = 0 \).

Key concepts

curl

Simply put, the curl of a vector field measures the rotation or spiraling tendency of the field around a point. In a 2D vector field, the curl will be a scalar value. For a vector field \(\textbf{F} = F_1 \mathbf{i} + F_2 \mathbf{j}\), the formula for the curl is \(abla \times \textbf{F} = \frac{\partial F_2}{\partial x} - \frac{\partial F_1}{\partial y}\). Here, we need to compute the partial derivatives: \(\frac{\partial F_2}{\partial x}\) and \(\frac{\partial F_1}{\partial y}\). For the vector field given, these partial derivatives are \(\frac{\partial y}{\partial x} = 0\) and \(\frac{\partial x}{\partial y} = 0\). As a result, the curl of this particular vector field is 0. If the curl is zero, the field has no rotation.

vector field

A vector field is an assignment of a vector to each point in a subset of space. Think of it as a function that associates a vector with every point in space. For example, in a 2D plane, a vector field can be visualized as arrows pointing in different directions and having different lengths, depending on their position. The given vector field is \(\textbf{r} = x \mathbf{i} + y \mathbf{j}\). This means at any point \((x, y)\), the vector has its components as the coordinates themselves: \(x\textbf{i}\textbf{j}y\textbf{j}\). Such a vector field might represent things like the flow of fluid in a plane or the influence of some force at every point in space. Understanding vector fields is crucial as they provide a comprehensive way to describe physical phenomena such as velocity fields in fluid dynamics, electromagnetic fields, and gravitational fields among others.