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. $$\mathbf{V}=x^{2} \mathbf{i}+y^{2} \mathbf{j}+z^{2} \mathbf{k}$$

Short answer

The divergence is \( 2x + 2y + 2z \) and the curl is \( \mathbf{0} \).

Step-by-step solution

Recall the Divergence Formula

The divergence of a vector field \( \mathbf{V} = V_x \mathbf{i} + V_y \mathbf{j} + V_z \mathbf{k} \) is given by \[ \text{div} \mathbf{V} = \frac{\partial V_x}{\partial x} + \frac{\partial V_y}{\partial y} + \frac{\partial V_z}{\partial z} \]

Compute the Partial Derivatives for Divergence

Given \( \mathbf{V} = x^{2} \mathbf{i} + y^{2} \mathbf{j} + z^{2} \mathbf{k} \), find the partial derivatives: \1. \frac{\partial}{x} x^2 = 2x \2. \frac{\partial}{y} y^2 = 2y \3. \frac{\partial}{z} z^2 = 2z

Calculate the Divergence

Using the divergence formula: \[ \text{div} \mathbf{V} = 2x + 2y + 2z \]

Recall the Curl Formula

The curl of a vector field \( \mathbf{V} = V_x \mathbf{i} + V_y \mathbf{j} + V_z \mathbf{k} \) is given by \[ \abla \times \mathbf{V} = \left( \frac{\partial V_z}{\partial y} - \frac{\partial V_y}{\partial z} \right) \mathbf{i} \ + \left( \frac{\partial V_x}{\partial z} - \frac{\partial V_z}{\partial x} \right) \mathbf{j} \ + \left( \frac{\partial V_y}{\partial x} - \frac{\partial V_x}{\partial y} \right) \mathbf{k} \]

Compute the Partial Derivatives for Curl

Given \( \mathbf{V} = x^{2} \mathbf{i} + y^{2} \mathbf{j} + z^{2} \mathbf{k} \), find the partial derivatives: \1. \frac{\partial z^2}{\partial y} = 0 \ and \ \frac{\partial y^2}{\partial z} = 0, \2. \frac{\partial x^2}{\partial z} = 0 \ and \ \frac{\partial z^2}{\partial x} = 0, \3. \frac{\partial y^2}{\partial x} = 0 \ and \ \frac{\partial x^2}{\partial y} = 0

Calculate the Curl

Using the curl formula: \[ \abla \times \mathbf{V} = (0 - 0) \mathbf{i} + (0 - 0) \mathbf{j} + (0 - 0) \mathbf{k} = \mathbf{0} \]

Key concepts

divergence

In vector calculus, the divergence of a vector field is a scalar measure of how much a vector field spreads out from a point. The divergence provides an intuitive sense of the 'flux' of the field. To calculate the divergence of a vector field \(\mathbf{V} = V_x \mathbf{i} + V_y \mathbf{j} + V_z \mathbf{k}\), we use the formula:
\[ \text{div} \mathbf{V} = \frac{\partial V_x}{\partial x} + \frac{\partial V_y}{\partial y} + \frac{\partial V_z}{\partial z} \]
Applying this to the given vector field \(\mathbf{V} = x^{2} \mathbf{i} + y^{2} \mathbf{j} + z^{2} \mathbf{k}\), we compute the partial derivatives:

• \(\frac{\partial}{\partial x} x^2 = 2x\)
• \(\frac{\partial}{\partial y} y^2 = 2y\)
• \(\frac{\partial}{\partial z} z^2 = 2z\)

Summing these, we get the divergence:
\[ \text{div} \mathbf{V} = 2x + 2y + 2z \] The divergence gives us insight into sources or sinks within the vector field, i.e., where the field is spreading out or converging.

curl

The curl of a vector field measures the rotation or swirling strength and direction of the field. Mathematically, the curl of a vector field \(\mathbf{V} = V_x \mathbf{i} + V_y \mathbf{j} + V_z \mathbf{k}\) is defined by:
\[ \abla \times \mathbf{V} = \left( \frac{\partial V_z}{\partial y} - \frac{\partial V_y}{\partial z} \right) \mathbf{i} + \left( \frac{\partial V_x}{\partial z} - \frac{\partial V_z}{\partial x} \right) \mathbf{j} + \left( \frac{\partial V_y}{\partial x} - \frac{\partial V_x}{\partial y} \right) \mathbf{k} \]
For our vector field \(\mathbf{V} = x^{2} \mathbf{i} + y^{2} \mathbf{j} + z^{2} \mathbf{k}\), we find the partial derivatives:

• \(\frac{\partial z^2}{\partial y} = 0\) and \(\frac{\partial y^2}{\partial z} = 0\)
• \(\frac{\partial x^2}{\partial z} = 0\) and \(\frac{\partial z^2}{\partial x} = 0\)
• \(\frac{\partial y^2}{\partial x} = 0\) and \(\frac{\partial x^2}{\partial y} = 0\)

Substituting these into the curl formula, we see that all terms are zero:
\[ \abla \times \mathbf{V} = (0 - 0) \mathbf{i} + (0 - 0) \mathbf{j} + (0 - 0) \mathbf{k} = \mathbf{0} \] Thus, the curl of this specific vector field is zero, indicating no rotational component.

partial derivatives

Partial derivatives are a fundamental concept in calculus, used extensively in vector calculus. They represent the rate of change of a function with respect to one variable while keeping other variables constant. This is crucial for understanding how functions behave in multi-variable contexts.

Consider a function \(f(x,y,z)\). The partial derivative with respect to \(x\) is written as \( \frac{\partial f}{\partial x}\) and indicates how \(f\) changes if \(x\) changes, holding \(y\) and \(z\) constant. Applying this to the given vector field components:

• For \(x^{2}\), the partial derivative with respect to \(x\) is \(2x\).
• For \(y^{2}\), the partial derivative with respect to \(y\) is \(2y\).
• For \(z^{2}\), the partial derivative with respect to \(z\) is \(2z\).

These derivatives help in calculating divergence and curl, as we've seen. They quantify how each component of the vector field changes with its respective variable, essential for deeper understanding and practical applications in physics and engineering.