Question

Find vector fields \(\mathbf{A}\) such that \(\mathbf{V}=\) curl \(\mathbf{A}\) for each given \(\mathbf{V}\) $$\mathbf{V}=\mathbf{i}\left(z e^{z y}+x \sin z x\right)+\mathbf{j} x \cos x z-\mathbf{k} z \sin z x$$

Short answer

An example solution is \( \textbf{A} = (\text{sin}(xz), -y \text{sin}(zx), -z e^{zy}) \)

Step-by-step solution

- Understand the Curl Operation

The curl of a vector field \(\textbf{A}\) is given by \(abla \times \textbf{A}\). We need to find a vector field \(\textbf{A} = A_x \textbf{i} + A_y \textbf{j} + A_z \textbf{k}\) where its curl equals the given vector field \(\textbf{V} = V_x \textbf{i} + V_y \textbf{j} - V_z \textbf{k}\).

- Compute the Components of Curl

For a vector field \(\textbf{A} = (A_x, A_y, A_z)\), compute its curl using the formula: \[ abla \times \textbf{A} = \begin{vmatrix} \textbf{i} & \textbf{j} & \textbf{k} \ \frac{\text{\textbackslash partial}}{\text{\textbackslash partial}x} & \frac{\text{\textbackslash partial}}{\text{\textbackslash partial}y} & \frac{\text{\textbackslash partial}}{\text{\textbackslash partial}z} \ A_x & A_y & A_z \ onumber \ \ onumber \ \end{vmatrix} \]

- Set Up the System of Partial Differential Equations

Given \(abla \times \textbf{A} = \textbf{i}(V_x) + \textbf{j}(V_y) + \textbf{k}(-V_z)\), we need to equate the components of the curl of \(\textbf{A}\) to the given \(\textbf{V} = \textbf{i}(ze^{zy} + x \text{sin}(zx)) + \textbf{j}(x \text{cos}(xz)) - \textbf{k}(z \text{sin}(zx))\). This gives us the equations: \(\frac{\text{\textbackslash partial}A_z}{\text{\textbackslash partial}y} - \frac{\text{\textbackslash partial}A_y}{\text{\textbackslash partial}z} = z e^{zy} + x \text{sin}(zx)\), \(\frac{\text{\textbackslash partial}A_x}{\text{\textbackslash partial}z} - \frac{\text{\textbackslash partial}A_z}{\text{\textbackslash partial}x} = x \text{cos}(xz)\), \(\frac{\text{\textbackslash partial}A_y}{\text{\textbackslash partial}x} - \frac{\text{\textbackslash partial}A_x}{\text{\textbackslash partial}y} = z \text{sin}(zx)\).

- Solve the System of Equations

Integrate the equations step by step. Firstly, integrate the second equation with respect to \(z\) to find \(A_x \): \[ \frac{\text{\textbackslash partial}A_x}{\text{\textbackslash partial}z} = x \text{cos}(xz) \ implies \ A_x = \text{sin}(xz) + f(x, y) \] Similarly, integrate the other equations and determine \(A_y\) and \(A_z\) by suitably choosing integration functions to satisfy all three components.

- Choose Consistent Potential Functions

Combine \(A_x\), \(A_y\), and \(A_z\) carefully to ensure the curl calculation matches the given \(\textbf{V}\). An example field could be \(\textbf{A} = (\text{sin}(xz), -y \text{sin}(zx), -z e^{zy})\). Test each partial derivative to confirm the curl matches \(\textbf{V}\).

- Verification

Verify our solution \[ abla \times \textbf{A} = (\frac{\text{\textbackslash partial}(-z e^{zy})}{\text{\textbackslash partial}y} - \frac{\text{\textbackslash partial}(-y \text{sin}(zx)}{\text{\textbackslash partial}z}, \ \frac{\text{\textbackslash partial}(\text{sin}(xz))}{\text{\textbackslash partial}z} - \frac{\text{\textbackslash partial}(-z e^{zy})}{\text{\textbackslash partial}x}, \ \frac{\text{\textbackslash partial}(-y \text{sin}(zx)}{\text{\textbackslash partial}x} - \frac{\text{\textbackslash partial}(\text{sin}(xz))}{\text{\textbackslash partial}y}) = (ze^{zy} + x \text{sin}(zx), x \text{cos}(xz), z \text{sin}(zx)) \]

Key concepts

Curl of a Vector Field

The curl of a vector field is a fundamental concept in vector calculus used to measure the rotation of a field. If you imagine fluid flow, the curl indicates how the fluid is rotating around a point. Mathematically, the curl of a vector field \(\mathbf{A}\) = \(A_x \mathbf{i} + A_y \mathbf{j} + A_z \mathbf{k}\) is given by \(abla \times \mathbf{A}\). This is represented as a determinant:\[abla \times \mathbf{A} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \ A_x & A_y & A_z \end{vmatrix}\]The result of this computation provides another vector field. Each component of the resulting vector is a combination of partial derivatives of the original vector field components. Knowing how to compute and interpret the curl can help you analyze physical phenomena, such as electromagnetic fields and fluid dynamics. To solve a problem where you are given a vector field and need to find another vector field whose curl matches the given one, you set up a system of partial differential equations corresponding to each component of the curl. The goal is to integrate these equations to find the original field.

Partial Differential Equations

Partial differential equations (PDEs) are equations that involve rates of change with respect to more than one variable. In vector calculus, PDEs often arise when working with operations like the curl or the divergence of vector fields. In our example, finding the vector field \(\mathbf{A}\) such that \(\mathbf{V} = abla \times \mathbf{A}\) involves solving a system of PDEs. Given the equations:\[ \frac{\partial A_z}{\partial y} - \frac{\partial A_y}{\partial z} = z e^{zy} + x \sin(zx) \]\[ \frac{\partial A_x}{\partial z} - \frac{\partial A_z}{\partial x} = x \cos(xz) \]\[ \frac{\partial A_y}{\partial x} - \frac{\partial A_x}{\partial y} = z \sin(zx) \]the challenge is to integrate these step by step. Start with one equation and integrate it with respect to one of the variables. For instance, integrating \(\frac{\partial A_x}{\partial z} = x \cos(xz)\) with respect to \(z\) gives \(A_x = \sin(xz) + f(x, y)\). Then, continue this process for the other equations, ensuring that any additional functions of integration appropriately match the other equations in the system. Successfully solving these requires careful attention to the conditions imposed by each equation.

Integration of Vector Fields

Integration of vector fields involves finding a vector field based on certain operations like curl or divergence applied to it. Given a vector field \(\mathbf{V}\), such that \(\mathbf{V} = abla \times \mathbf{A}\), we use integration to determine \(\mathbf{A}\). This process involves following a systematic approach:

• Compute individual components: Solve each partial differential equation derived from comparing the curl of \(\mathbf{A}\) with the given \(\mathbf{V}\).
  
• Integrate step by step: Begin with integrating one component, then use results to inform subsequent integrations.
  
• Combine results: Ensure the resulting components form a consistent vector field that matches the original requirements.
  

By integrating each part of the system carefully, we derive the components \(A_x, A_y, A_z\). For example, we integrated \(\frac{\partial A_x}{\partial z} = x \cos(xz)\) to find \(A_x = \sin(xz) + f(x, y)\). Then, integrating other equations similarly, we ensure that the full vector field \(\mathbf{A}\) corrects to match the curl \(\mathbf{V}\). Practical applications of these techniques include solving physical problems in electromagnetism, where one often needs to find potential fields from known forces or fields.