https://file.aiquickdraw.com/mathbot/file/1731788119899x7tu3zz4.jpg

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:
\[
\nabla \times \mathbf{F} =
\begin{vmatrix}
\mathbf{i} & \mathbf{j} & \mathbf{k} \\
\frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\
\sin(x - y) & \sin(y - z) & \sin(z - x)
\end{vmatrix}
\]

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.

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.

Learn how to compute the curl of the vector field F(x, y, z) = sin(x - y)i + sin(y - z)j + sin(z - x)k. This solution includes detailed steps using vector calculus principles, the curl determinant formula, and partial derivatives. Ideal for undergraduate students studying vector calculus.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation