Compute the Curl of a Vector Field F(x, y, z) with Sinusoidal Components

Math Problem Statement

Use a computer algebra system to find the curl F for the vector field F(x, y, z) = sin(x - y)i + sin(y - z)j + sin(z - x)k.

Solution

To compute the curl of the vector field $\mathbf{F}(x, y, z) = \sin(x - y)\mathbf{i} + \sin(y - z)\mathbf{j} + \sin(z - x)\mathbf{k}$ , we use the curl formula:

$\text{curl} \, \mathbf{F} = \nabla \times \mathbf{F}$

In component form:

\begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ \sin(x - y) & \sin(y - z) & \sin(z - x) \end{vmatrix}$$ Let's compute this step by step: 1. **Expand the determinant**: $$\text{curl} \, \mathbf{F} = \mathbf{i} \left( \frac{\partial}{\partial y} \sin(z - x) - \frac{\partial}{\partial z} \sin(y - z) \right) - \mathbf{j} \left( \frac{\partial}{\partial x} \sin(z - x) - \frac{\partial}{\partial z} \sin(x - y) \right) + \mathbf{k} \left( \frac{\partial}{\partial x} \sin(y - z) - \frac{\partial}{\partial y} \sin(x - y) \right)$$ 2. **Compute partial derivatives**: - $$\frac{\partial}{\partial y} \sin(z - x) = 0$$ (no $$y$$-dependence). - $$\frac{\partial}{\partial z} \sin(y - z) = -\cos(y - z)$$. - $$\frac{\partial}{\partial x} \sin(z - x) = -\cos(z - x)$$. - $$\frac{\partial}{\partial z} \sin(x - y) = 0$$ (no $$z$$-dependence). - $$\frac{\partial}{\partial x} \sin(y - z) = 0$$ (no $$x$$-dependence). - $$\frac{\partial}{\partial y} \sin(x - y) = -\cos(x - y)$$. 3. **Substitute into the determinant**: $$\text{curl} \, \mathbf{F} = \mathbf{i} \left( 0 - (-\cos(y - z)) \right) - \mathbf{j} \left( -\cos(z - x) - 0 \right) + \mathbf{k} \left( 0 - (-\cos(x - y)) \right)$$ Simplify each term: - $$\mathbf{i}$$-component: $$\cos(y - z)$$, - $$\mathbf{j}$$-component: $$-(-\cos(z - x)) = \cos(z - x)$$, - $$\mathbf{k}$$-component: $$-(-\cos(x - y)) = \cos(x - y)$$. 4. **Write the final result**: $$\text{curl} \, \mathbf{F} = \cos(y - z)\mathbf{i} + \cos(z - x)\mathbf{j} + \cos(x - y)\mathbf{k}.$$ --- Would you like further clarification or a visualization of the curl? Let me know! 😊 Here are 5 related questions you might consider: 1. How does the curl relate to the physical interpretation of a vector field? 2. What does it mean if the curl of a vector field is zero? 3. How would the computation change if $$\mathbf{F}$$ had additional components? 4. How can the divergence of $$\mathbf{F}$$ be calculated, and how is it different from the curl? 5. What applications in physics use the concept of curl? **Tip:** Remember that the curl measures the rotational tendency of a vector field, often visualized as "vorticity" in fluid dynamics or electromagnetism.

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Math Problem Analysis

Mathematical Concepts

Vector Calculus
Curl of a Vector Field
Partial Derivatives

Formulas

Curl formula: ∇ × F
Determinant for curl: ∇ × F = |i j k| / |∂/∂x ∂/∂y ∂/∂z| / |F₁ F₂ F₃|

Theorems

Stokes' Theorem (contextually relevant for curl)

Suitable Grade Level

Undergraduate Mathematics or Physics

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