Question

Compute the divergence and the curl of each of the following vector fields.\(\mathrm{V}=y_{i}+z \mathrm{j}+x \mathrm{k}\)

Short answer

The divergence of the vector field is 0. The curl of the vector field is \(- \mathbf{i} - \mathbf{j} + \mathbf{k}\).

Step-by-step solution

Write down the given vector field

The given vector field is \(\mathbf{V} = yi + zj + xk \).

Compute the divergence of the vector field

The divergence of a vector field \( \mathbf{V} = P \mathbf{i} + Q \mathbf{j} + R \mathbf{k} \) is given by the formula: \(abla \cdot \mathbf{V} = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} + \frac{\partial R}{\partial z} \).In our case, \( P = y \), \( Q = z \) and \( R = x \).Compute each partial derivative term:\( \frac{\partial y}{\partial x} = 0 \), \( \frac{\partial z}{\partial y} = 0 \), and \( \frac{\partial x}{\partial z} = 0 \).Sum these results to get the divergence:\( abla \cdot \mathbf{V} = 0 + 0 + 0 = 0 \).

Compute the curl of the vector field

The curl of a vector field \( \mathbf{V} = P \mathbf{i} + Q \mathbf{j} + R \mathbf{k} \) is given by the formula:\[ abla \times \mathbf{V} = \begin{vmatrix}\mathbf{i} & \mathbf{j} & \mathbf{k} \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \P & Q & R \end{vmatrix}\].Substitute \( P = y \), \( Q = z \), and \( R = x \) into the determinant:\[ abla \times \mathbf{V} = \begin{vmatrix}\mathbf{i} & \mathbf{j} & \mathbf{k} \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \ y & z & x \end{vmatrix}\].Calculate the determinant by finding the component-wise partial derivatives:\[ \mathbf{i} \left( \frac{\partial x}{\partial y} - \frac{\partial z}{\partial z} \right) - \mathbf{j} \left( \frac{\partial x}{\partial x} - \frac{\partial y}{\partial z} \right) + \mathbf{k} \left( \frac{\partial y}{\partial y} - \frac{\partial z}{\partial x} \right) \]Simplifying each term, we get:\[ \mathbf{i} (0 - 1) - \mathbf{j} (1 - 0) + \mathbf{k} (1 - 0) = - \mathbf{i} - \mathbf{j} + \mathbf{k}\].Thus, the curl of the vector field is \(abla \times \mathbf{V} = - \mathbf{i} - \mathbf{j} + \mathbf{k} \).

Key concepts

Divergence

Divergence is a scalar value that measures how much a vector field spreads out from a point. Think of it like measuring the flow of water from a source. For a vector field \(\mathbf{V} = P \mathbf{i} + Q \mathbf{j} + R \mathbf{k}\), the divergence is computed as: \[ abla \cdot \mathbf{V} = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} + \frac{\partial R}{\partial z} \]. If the divergence is zero, the flow is neither expanding nor compressing at that point. In the provided exercise, the vector field \(\mathbf{V} = yi + zj + xk\) has zero divergence because: \[ \frac{\partial y}{\partial x} = 0, \frac{\partial z}{\partial y} = 0, \frac{\partial x}{\partial z} = 0, \quad \text{thus } abla \cdot \mathbf{V} = 0 + 0 + 0 = 0 \].

Curl

Curl measures the rotation or the swirling strength of a vector field around a point. Imagine how a whirlpool would look in water. The curl of a vector field \(\mathbf{V} = P \mathbf{i} + Q \mathbf{j} + R \mathbf{k}\) can be found using the determinant: \[ abla \times \mathbf{V} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \ P & Q & R \end{vmatrix} \]. For the given field \(\mathbf{V} = yi + zj + xk\), this determinant expands to: \[ \mathbf{i}(0 - 1) - \mathbf{j}(1- 0) + \mathbf{k}(1 - 0) = -\mathbf{i} - \mathbf{j} + \mathbf{k} \]. Therefore, the curl is \(abla \times \mathbf{V} = -\mathbf{i} - \mathbf{j} + \mathbf{k}\).

Vector Field

A vector field assigns a vector to every point in space. Imagine wind represented in a weather map, where every point has a direction and magnitude. The given field \(\mathbf{V} = yi + zj + xk\) means that at each point in space, the vector values change depending on the coordinates. In this vector field: \[ P = y, \ Q = z, \ R = x \]. It helps interpret physical situations like fluid flow, electromagnetism, and more. Understanding how the field behaves involves tools like divergence and curl.

Partial Derivatives

Partial derivatives measure how a function changes as one variable changes while keeping others constant. Imagine you have a function \(f(x, y, z)\). The partial derivative \(\frac{\partial f}{\partial x}\) tells us how \(f\) changes as only \(x\) changes, with \(y\) and \(z\) fixed. This is crucial in vector calculus as it helps to compute divergence and curl. For the vector field \(\mathbf{V} = yi + zj + xk\), the derivatives are: \[ \frac{\partial y}{\partial x} = 0, \frac{\partial z}{\partial y} = 0, \frac{\partial x}{\partial z} = 0 \], making it easier to find divergence and curl. Analyzing these changes improves our understanding of the field's behavior.