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

Short answer

Divergence: \(2 + x \sinh z\), Curl: 0

Step-by-step solution

- Understanding Divergence

The divergence of a vector field \[ \mathbf{V} = P \mathbf{i} + Q \mathbf{j} + R \mathbf{k} \] is given by the formula: \[ abla \bullet \textbf{V} = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} + \frac{\partial R}{\partial z} \]

- Compute Partial Derivatives for Divergence

Given \[ \mathbf{V} = \sinh z \mathbf{i} + 2 y \mathbf{j} + x \cosh z \mathbf{k} \], compute the partial derivatives: \[ \frac{\partial P}{\partial x} = \frac{\partial (\sinh z)}{\partial x} = 0\] \[ \frac{\partial Q}{\partial y} = \frac{\partial (2 y)}{\partial y} = 2\] \[ \frac{\partial R}{\partial z} = \frac{\partial (x \cosh z)}{\partial z} = x \sinh z \]

- Calculate Divergence

Add the partial derivatives: \[ abla \bullet \mathbf{V} = 0 + 2 + x \sinh z = 2 + x \sinh z \]

- Understanding Curl

The curl of a vector field \[ \mathbf{V} = P \mathbf{i} + Q \mathbf{j} + R \mathbf{k} \] is given by the formula: \[ abla \times \mathbf{V} = \left( \frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z} \right) \mathbf{i} + \left( \frac{\partial P}{\partial z} - \frac{\partial R}{\partial x} \right) \mathbf{j} + \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) \mathbf{k} \]

- Compute Partial Derivatives for Curl

Compute the partial derivatives: \[ \frac{\partial R}{\partial y} = \frac{\partial (x \cosh z)}{\partial y} = 0 \] \[ \frac{\partial Q}{\partial z} = \frac{\partial (2 y)}{\partial z} = 0 \] \[ \frac{\partial P}{\partial z} = \frac{\partial (\sinh z)}{\partial z} = \cosh z \] \[ \frac{\partial R}{\partial x} = \frac{\partial (x \cosh z)}{\partial x} = \cosh z \] \[ \frac{\partial Q}{\partial x} = \frac{\partial (2 y)}{\partial x} = 0 \] \[ \frac{\partial P}{\partial y} = \frac{\partial (\sinh z)}{\partial y} = 0 \]

- Calculate Curl

Substitute the derived partials into the curl formula: \[ abla \times \mathbf{V} = (0 - 0) \mathbf{i} + (\cosh z - \cosh z) \mathbf{j} + (0 - 0) \mathbf{k} = 0 \]

Key concepts

divergence

Divergence is a measure of the rate at which a vector field spreads out from a point. Think of it as how much a vector field 'diverges' or moves away from a given point. It's calculated using partial derivatives: if you have a vector field \(\textbf{V} = P \textbf{i} + Q \textbf{j} + R \textbf{k} \), the divergence formula is given by \(abla \bullet \textbf{V} = \frac{\frac\partial P\frac\partial x} + \frac{\frac\partial Q\frac\partial y} + \frac{\frac\partial R\frac\partial z} \).

Here's how you apply the formula:

• Identify the components of the vector field: P, Q, and R.
• Find the partial derivatives of these components with respect to x, y, and z.
• Sum up the partial derivatives.

For the given vector field, \(\textbf{V} = \text{sinh} z \textbf{i} + 2 y \textbf{j} + x \text{cosh} z \textbf{k} \), the partial derivatives are: \(\frac{\frac\partial (\text{sinh} z)\frac\partial x} = 0 \), \(\frac{\frac\partial (2y)\frac\partial y} = 2 \), and \(\frac{\frac\partial (x \text{cosh} z)\frac\partial z} = x \text{sinh} z \).
Adding them gives us: \(abla \bullet \textbf{V} = 0 + 2 + x \text{sinh} z = 2 + x \text{sinh} z \).

curl

Curl measures the rotation of a vector field. Imagine placing a small paddle wheel in the flow of the vector field; the curl would tell you how much the wheel tends to rotate. The formula for curl is \(abla \times \textbf{V} = \begin{vmatrix} \textbf{i} & \textbf{j} & \textbf{k} \ \frac{\frac\partial\frac\partial x} & \frac{\frac\partial\frac\partial y} & \frac{\frac\partial\frac\partial z} \ P & Q & R \end{vmatrix} \).

Steps to compute the curl:

• Set up the determinant involving unit vectors i, j, k.
• Compute the required partial derivatives.
• Plug in these values and solve the determinant.

For our vector field \(\textbf{V} = \text{sinh} z \textbf{i} + 2 y \textbf{j} + x \text{cosh} z \textbf{k} \):

• Partial derivatives: \(\frac{\frac\partial (2y)\frac\partial z} = 0 \), \(\frac{\frac\partial (\text{sinh} z)\frac\partial z} = \text{cosh} z \), \(\frac{\frac\partial (x \text{cosh} z)\frac\partial y} = 0 \).
• Using these, the curl formula simplifies to: \(abla \times \textbf{V} = (0-0) \textbf{i} + (\text{cosh} z - \text{cosh} z) \textbf{j} + (0-0) \textbf{k} = 0 \).

partial derivatives

Partial derivatives are the building blocks of vector calculus. They tell us how a function changes as each individual variable changes while other variables remain constant.

When dealing with vector fields, you'll often see notations like \(\frac{\frac\partial P\frac\partial x} \) which means 'the partial derivative of P with respect to x.'

Here's how to find them:

• Identify the variable you're differentiating with respect to.
• Differentiate while treating all other variables as constants.

In the vector field \(\textbf{V} = \text{sinh} z \textbf{i} + 2 y \textbf{j} + x \text{cosh} z \textbf{k} \), we found: \(\frac{\frac\partial (\text{sinh} z)\frac\partial x} = 0 \), \(\frac{\frac\partial (2 y)\frac\partial y} = 2 \), and \(\frac{\frac\partial (x \text{cosh} z)\frac\partial z} = x \text{sinh} z \).

vector fields

A vector field assigns a vector to every point in space. Imagine each point in space having its own arrow indicating a direction and magnitude. Examples include magnetic fields, fluid flow, and wind patterns.

For example, the vector field \(\textbf{V} = \text{sinh} z \textbf{i} + 2 y \textbf{j} + x \text{cosh} z \textbf{k} \) specifies a vector at every point \((x, y, z) \).

How to work with vector fields:

• Identify the components: P, Q, and R.
• Use these components to compute divergence and curl.
• Visualize the field to understand its behavior in space.

gradient operators

Gradient operators are used to find the gradient, divergence, and curl of vector fields. \(abla \) is called the del operator and it's a central tool in vector calculus.

The del operator is expressed as \(abla = \frac{\frac\partial\frac\partial x} \textbf{i} + \frac{\frac\partial\frac\partial y} \textbf{j} + \frac{\frac\partial\frac\partial z} \textbf{k} \).

Uses of the gradient operator:

• Calculates the gradient of a scalar field: it gives the direction and rate of fastest increase.
• Calculates divergence: gives a scalar telling how much a vector field spreads out.
• Calculates curl: gives a vector representing rotational circulation at a point.

For example, to find the divergence and curl of the given vector field \(\textbf{V} = \text{sinh} z \textbf{i} + 2 y \textbf{j} + x \text{cosh} z \textbf{k} \), we applied \(abla \) to compute partial derivatives and solve using our known formulas.