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. $$\mathbf{V}=z \mathbf{i}+y \mathbf{j}+x \mathbf{k}$$

Short answer

Divergence: 1. Curl: -i - j.

Step-by-step solution

Understand the Vector Field

The given vector field is \(\textbf{V} = z \textbf{i} + y \textbf{j} + x \textbf{k}\). This means the x-component is z, the y-component is y, and the z-component is x.

Compute the Divergence

The divergence of a vector field \(\textbf{V} = P \textbf{i} + Q \textbf{j} + R \textbf{k}\) is given by \(abla \bullet \textbf{V} = \frac{\text{∂}P}{\text{∂}x} + \frac{\text{∂}Q}{\text{∂}y} + \frac{\text{∂}R}{\text{∂}z}\). For our vector field, \(P = z, Q = y, R = x\). So calculate: \(abla \bullet \textbf{V} = \frac{\text{∂}z}{\text{∂}x} + \frac{\text{∂}y}{\text{∂}y} + \frac{\text{∂}x}{\text{∂}z} = 0 + 1 + 0 = 1\).

Set Up and Calculate the Curl

The curl of a vector field \(\textbf{V} = P \textbf{i} + Q \textbf{j} + R \textbf{k}\) is given by \(abla \times \textbf{V} = \begin{vmatrix} \textbf{i} & \textbf{j} & \textbf{k} \ \frac{\text{∂}}{\text{∂}x} & \frac{\text{∂}}{\text{∂}y} & \frac{\text{∂}}{\text{∂}z} \ P & Q & R \ \end{vmatrix}\). For our vector field, we need to calculate: \(abla \times \textbf{V} = \begin{vmatrix} \textbf{i} & \textbf{j} & \textbf{k} \ \frac{\text{∂}}{\text{∂}x} & \frac{\text{∂}}{\text{∂}y} & \frac{\text{∂}}{\text{∂}z} \ z & y & x \ \end{vmatrix}\). The expanded form is: \ \textbf{i} \bigg( \frac{\text{∂}x}{\text{∂}y} - \frac{\text{∂}y}{\text{∂}z} \bigg) - \textbf{j} \bigg( \frac{\text{∂}x}{\text{∂}x} - \frac{\text{∂}z}{\text{∂}z} \bigg) + \textbf{k} \bigg( \frac{\text{∂}y}{\text{∂}x} - \frac{\text{∂}z}{\text{∂}y} \bigg) = \textbf{i}(0 - 1) - \textbf{j}(1 - 0) + \textbf{k}(0 - 0) = -\textbf{i} - \textbf{j}.

Key concepts

Vector Fields

A vector field is a function that assigns a vector to each point in space. Imagine a weather map showing wind speed and direction at various locations; this is a real-life example of a vector field. In the given problem, the vector field is defined as \(\textbf{V} = z \textbf{i} + y \textbf{j} + x \textbf{k} \). Here, \(\textbf{V} = z \textbf{i} + y \textbf{j} + x \textbf{k}\) means that at any point in space, the vector has: \ - a component z in the x-direction, represented by the unit vector \textbf{i}\ \- a component y in the y-direction, represented by the unit vector \textbf{j}\ \- a component x in the z-direction, represented by the unit vector \textbf{k}\ Understanding vector fields helps us describe physical phenomena where direction and magnitude are essential, such as electromagnetic fields or fluid flows.

Divergence

Divergence of a vector field measures the 'outflowing-ness' at a given point; it indicates how much a vector field spreads out from a particular point. Mathematically, for a vector field \(\textbf{V} = P \textbf{i} + Q \textbf{j} + R \textbf{k}\), the divergence is given by \(abla \bullet \textbf{V} = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} + \frac{\partial R}{\partial z}\)\. For the given vector field \(\textbf{V} = z \textbf{i} + y \textbf{j} + x \textbf{k}\), let \P = z, Q = y, R = x\. Computing the divergence:\(\abla \bullet \textbf{V} = \frac{\partial z}{\partial x} + \frac{\partial y}{\partial y} + \frac{\partial x}{\partial z} = 0 + 1 + 0 = 1\). The result shows that the divergence is \(abla \bullet \textbf{V} = 1\). Reaching a non-zero divergence indicates that there is a net 'outflow' at each point in the vector field.

Curl

Curl of a vector field measures the rotational tendency at a given point, indicating how much and in which direction the field 'twists' or 'rotates' around that point. For a vector field \(\textbf{V} = P \textbf{i} + Q \textbf{j} + R \textbf{k} \), the curl is computed as the determinant: \[\abla \times \textbf{V} = \begin{vmatrix} \textbf{i} & \textbf{j} & \textbf{k} \ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \ P & Q & R \end{vmatrix} \]. For our vector field \( \textbf{V} = z \textbf{i} + y \textbf{j} + x \textbf{k} \), we calculate: \[\abla \times \textbf{V} = \begin{vmatrix} \textbf{i} & \textbf{j} & \textbf{k} \ \partial x\partial x {{\partial x}} \ y & x & z & y & z \ \end{vmatrix} \]\. Expanded, this reads: \[\textbf{i} \big(\frac{\partial x}{\partial y} - \frac{\partial y}{\partial z}\big) - \ \big(\frac{\partial x}{\partial x} - \frac{\partial z}{\partial z}\big) + \ \big(\frac{\partial y}{\partial x} - \frac{\partial z}{\partial y} \big) = \textbf{i}(0 - 1) + {\partial y}{\partial z} = -\textbf{./i} - \textbf{j} \]. The curl of this vector field shows -1 in the \( \textbf{i} \) and \( \textbf{j} \) directions, indicating a rotational tendency in those planes. Understanding divergence and curl alongside vector fields can illuminate the behavior and properties of physical systems, from fluid flow to electromagnetic fields, providing a deeper grasp of the underlying principles. Always practice these calculations by hand first, to strengthen your foundational knowledge.