Math Problem Statement

Consider the vector field F defined in cylindrical coordinates (r, θ, z) by: F(r, θ, z) = rer + zeθ + erez where er, eθ, and ez are the unit vectors in the r, θ, and z directions, respectively. For example, eθ points into the direction of the increase of θ, and has unit length. Find the curl of this vector field, and express the answer back in the cylindrical coordinates.

Solution

To compute the curl of a vector field in cylindrical coordinates, we use the general formula:

\begin{vmatrix} \mathbf{e_r} & \frac{1}{r}\mathbf{e_\theta} & \mathbf{e_z} \\ \frac{\partial}{\partial r} & \frac{1}{r} \frac{\partial}{\partial \theta} & \frac{\partial}{\partial z} \\ F_r & r F_\theta & F_z \end{vmatrix}$$ Here: - $$\mathbf{F} = F_r \mathbf{e_r} + F_\theta \mathbf{e_\theta} + F_z \mathbf{e_z}$$, - $$F_r = r$$, $$F_\theta = z$$, $$F_z = e^r$$. ### Step 1: Substitute the components into the determinant $$\text{curl} \, \mathbf{F} = \begin{vmatrix} \mathbf{e_r} & \frac{1}{r}\mathbf{e_\theta} & \mathbf{e_z} \\ \frac{\partial}{\partial r} & \frac{1}{r} \frac{\partial}{\partial \theta} & \frac{\partial}{\partial z} \\ r & rz & e^r \end{vmatrix}$$ ### Step 2: Expand the determinant We compute each component separately. The curl is given by: $$\text{curl} \, \mathbf{F} = \left( \frac{\partial (r z)}{\partial z} - \frac{\partial (e^r)}{\partial \theta} \right) \mathbf{e_r} + \left( \frac{\partial (e^r)}{\partial r} - \frac{\partial r}{\partial z} \right) \frac{\mathbf{e_\theta}}{r} + \left( \frac{\partial (rz)}{\partial r} - \frac{1}{r} \frac{\partial (r)}{\partial \theta} \right) \mathbf{e_z}.$$ #### Compute each term: 1. **Radial component ($$\mathbf{e_r}$$):** $$\frac{\partial (rz)}{\partial z} - \frac{\partial (e^r)}{\partial \theta} = r - 0 = r.$$ 2. **Azimuthal component ($$\frac{\mathbf{e_\theta}}{r}$$):** $$\frac{\partial (e^r)}{\partial r} - \frac{\partial r}{\partial z} = e^r - 0 = e^r.$$ Multiply by $$\frac{1}{r}$$: $$\frac{e^r}{r}.$$ 3. **Axial component ($$\mathbf{e_z}$$):** $$\frac{\partial (rz)}{\partial r} - \frac{1}{r} \frac{\partial (r)}{\partial \theta} = z - 0 = z.$$ ### Step 3: Combine the components $$\text{curl} \, \mathbf{F} = r \mathbf{e_r} + \frac{e^r}{r} \mathbf{e_\theta} + z \mathbf{e_z}.$$ ### Final Answer: $$\boxed{\nabla \times \mathbf{F} = r \mathbf{e_r} + \frac{e^r}{r} \mathbf{e_\theta} + z \mathbf{e_z}}$$ This is the curl of the vector field $$\mathbf{F}$$ expressed in cylindrical coordinates. --- Would you like me to verify any steps or explain further details? Here are 5 follow-up questions you might explore: 1. What is the divergence of $$\mathbf{F}$$ in cylindrical coordinates? 2. How does the curl change in Cartesian coordinates? 3. How would the result differ if the components of $$\mathbf{F}$$ included $$\theta$$-dependent terms? 4. Can the vector field $$\mathbf{F}$$ be expressed as the gradient of a scalar field? 5. How does the physical interpretation of curl apply to this field? **Tip:** Always verify your result by checking whether $$\nabla \cdot (\nabla \times \mathbf{F}) = 0$$, which is a property of the curl in any coordinate system.

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Math Problem Analysis

Mathematical Concepts

Vector Calculus
Cylindrical Coordinates
Curl of a Vector Field

Formulas

Curl in cylindrical coordinates: ∇ × F = |er eθ/r ez ; ∂/∂r (1/r)∂/∂θ ∂/∂z ; Fr rFθ Fz|
Component-wise calculation for curl: (∂/∂y Fz - ∂/∂z Fy) etc.

Theorems

Properties of curl
Determinant method for computing vector operations

Suitable Grade Level

Undergraduate Mathematics or Engineering

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