Question

Compute the divergence and the curl of each of the following vector fields.\(V=x^{2} i+y^{2} \mathbf{j}+z^{2} k\)

Short answer

Divergence: 2x + 2y + 2z, Curl: 0

Step-by-step solution

- Compute the Divergence

The divergence of a vector field \( \textbf{V} = V_x \textbf{i} + V_y \textbf{j} + V_z \textbf{k} \) is given by \( abla \bullet \textbf{V} = \frac{\text{d}V_x}{\text{d}x} + \frac{\text{d}V_y}{\text{d}y} + \frac{\text{d}V_z}{\text{d}z} \). For the vector field \( \textbf{V} = x^2 \textbf{i} + y^2 \textbf{j} + z^2 \textbf{k} \), compute the partial derivatives: \(\frac{\text{d}(x^2)}{\text{d}x} = 2x\), \(\frac{\text{d}(y^2)}{\text{d}y} = 2y\), and \(\frac{\text{d}(z^2)}{\text{d}z} = 2z\). Summing these partial derivatives gives \(abla \bullet \textbf{V} = 2x + 2y + 2z \).

- Compute the Curl

The curl of a vector field \( \textbf{V} = V_x \textbf{i} + V_y \textbf{j} + V_z \textbf{k} \) is given by \( abla \times \textbf{V} \), which we can write as a determinant: \[ abla \times \textbf{V} = \begin{vmatrix} \textbf{i} & \textbf{j} & \textbf{k} \ \frac{\text{d}}{\text{d}x} & \frac{\text{d}}{\text{d}y} & \frac{\text{d}}{\text{d}z} \ V_x & V_y & V_z \ \end{vmatrix} \] For \( \textbf{V} = x^2 \textbf{i} + y^2 \textbf{j} + z^2 \textbf{k} \), this determinant becomes: \[ abla \times \textbf{V} = \begin{vmatrix} \textbf{i} & \textbf{j} & \textbf{k} \ \frac{\text{d}}{\text{d}x} & \frac{\text{d}}{\text{d}y} & \frac{\text{d}}{\text{d}z} \ x^2 & y^2 & z^2 \ \end{vmatrix} = (\frac{\text{d}z^2}{\text{d}y} - \frac{\text{d}y^2}{\text{d}z})\textbf{i} - (\frac{\text{d}z^2}{\text{d}x} - \frac{\text{d}x^2}{\text{d}z})\textbf{j} + (\frac{\text{d}y^2}{\text{d}x} - \frac{\text{d}x^2}{\text{d}y})\textbf{k} \] Computing these partial derivatives, we get: \ \frac{\text{d}z^2}{\text{d}y} = 0 \frac{\text{d}y^2}{\text{d}z} = 0 \frac{\text{d}z^2}{\text{d}x} = 0 \frac{\text{d}x^2}{\text{d}z} = 0 \frac{\text{d}y^2}{\text{d}x} = 0 \frac{\text{d}x^2}{\text{d}y} = 0 \. Hence, \(abla \times \textbf{V} = 0\textbf{i} - 0\textbf{j} + 0\textbf{k} = \textbf{0} \).

Key concepts

Divergence of a Vector Field

Divergence is a measure of how much a vector field expands or contracts at a given point. You can think of it like air flowing out of a balloon. If the air flows out evenly, it spreads in all directions.
Mathematically, the divergence of a vector field \( \textbf{V} = V_x \textbf{i} + V_y \textbf{j} + V_z \textbf{k} \) is calculated as the sum of the partial derivatives of each component:
\(abla \bullet \textbf{V} = \frac{\text{d}V_x}{\text{d}x} + \frac{\text{d}V_y}{\text{d}y} + \frac{\text{d}V_z}{\text{d}z} \).
In our example, for \(\textbf{V} = x^2 \textbf{i} + y^2 \textbf{j} + z^2 \textbf{k}\), we compute:
\(\frac{\text{d}(x^2)}{\text{d}x} = 2x\text{, } \frac{\text{d}(y^2)}{\text{d}y} = 2y\text{, } \frac{\text{d}(z^2)}{\text{d}z} = 2z\). So, \(abla \bullet \textbf{V} = 2x + 2y + 2z\). This value tells us how much the field is spreading out or converging at each point in space.

Curl of a Vector Field

Curl measures the rotation or twisting of a vector field around a point, like water turning around a point in a whirlpool. Mathematically, the curl of a vector field \(\textbf{V} = V_x \textbf{i} + V_y \textbf{j} + V_z \textbf{k}\) is found using a determinant:
\(abla \times \textbf{V} = \begin{vmatrix} \textbf{i} & \textbf{j} & \textbf{k} \ \frac{\text{d}}{\text{d}x} & \frac{\text{d}}{\text{d}y} & \frac{\text{d}}{\text{d}z} \ V_x & V_y & V_z \ \end{vmatrix} \).
For \(\textbf{V} = x^2 \textbf{i} + y^2 \textbf{j} + z^2 \textbf{k}\), this gives: \(abla \times \textbf{V} = \begin{vmatrix} \textbf{i} & \textbf{j} & \textbf{k} \ \frac{\text{d}}{\text{d}x} & \frac{\text{d}}{\text{d}y} & \frac{\text{d}}{\text{d}z} \ x^2 & y^2 & z^2 \ \end{vmatrix} \),
which simplifies to
abla \times \textbf{V} = (\frac{\text{d}z^2}{\text{d}y} - \frac{\text{d}y^2}{\text{d}z})\textbf{i} - (\frac{\text{d}z^2}{\text{d}x} - \frac{\text{d}x^2}{\text{d}z})\textbf{j} + (\frac{\text{d}y^2}{\text{d}x} - \frac{\text{d}x^2}{\text{d}y})\textbf{k} .
In simpler terms, all these partial derivatives are zero, because squaring a variable removes dependence on others: \(\frac{\text{d}z^2}{\text{d}y} = 0 \text{, } \frac{\text{d}y^2}{\text{d}z} = 0 \text{, } \frac{\text{d}z^2}{\text{d}x} = 0 \text{, } \frac{\text{d}x^2}{\text{d}z} = 0 \text{, } \frac{\text{d}y^2}{\text{d}x} = 0 \text{, } \frac{\text{d}x^2}{\text{d}y} = 0\). Consequently, \(abla \times \textbf{V} = \textbf{0} \), meaning there is no rotation in this vector field.

What Is a Vector Field?

A vector field is a function that assigns a vector to each point in space. Imagine you have an arrow—a vector—at every point in a region of space, each of which could represent quantities like wind speed or magnetic force.
In this exercise, our vector field \(\textbf{V} = x^2 \textbf{i} + y^2 \textbf{j} + z^2 \textbf{k}\) assigns a vector \((x^2, y^2, z^2)\) to every point \((x, y, z)\) in space.

Examples of vector fields include:

• Weather maps showing wind speed and direction at different locations
• Gravitational fields showing how gravitational force varies around a mass

Understanding vector fields allows us to model and analyze physical phenomena in engineering, physics, and other sciences.
By computing quantities like divergence and curl, we can better understand the behavior and properties of these fields.