Problem 31 - Thomas Calculus

Question

Find a vector field with twice-differentiable components whose curl is \(x \mathbf{i}+y \mathbf{j}+z \mathbf{k}\) or prove that no such field exists.

Step-by-Step Solution

Answer

No such vector field exists since the curl vector is not divergence-free.

Step 1: Understand the question

We need to find whether there exists a vector field \( \mathbf{F} = P \mathbf{i} + Q \mathbf{j} + R \mathbf{k} \) such that the curl of \( \mathbf{F} \) equals \( x \mathbf{i} + y \mathbf{j} + z \mathbf{k} \). The curl of \( \mathbf{F} \) is given by \( abla \times \mathbf{F} = \left( \frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z} \right) \mathbf{i} + \left( \frac{\partial P}{\partial z} - \frac{\partial R}{\partial x} \right) \mathbf{j} + \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) \mathbf{k} \).

Step 2: Apply the conditions for the existence of curl

For a vector field \( \mathbf{F} \) to have a curl \( \mathbf{C} = x \mathbf{i} + y \mathbf{j} + z \mathbf{k} \), the vector \( \mathbf{C} \) must be divergence-free. This means \( abla \cdot \mathbf{C} = 0 \). Compute this divergence: \( abla \cdot \mathbf{C} = \frac{\partial}{\partial x}(x) + \frac{\partial}{\partial y}(y) + \frac{\partial}{\partial z}(z) = 1+1+1 = 3 eq 0 \).

Step 3: Interpret the result

Since \( abla \cdot \mathbf{C} eq 0 \), \( \mathbf{C} = x \mathbf{i} + y \mathbf{j} + z \mathbf{k} \) cannot be a curl of any vector field. For a vector field to exist, the curl \( \mathbf{C} \) must be divergence-free, which is not satisfied in this case.

Key Concepts

Curl

When discussing vector fields, the curl is a measure of the field's rotation at a point. Essentially, the curl tells us how fast and in what direction the field is "twisting". Consider a vector field \( \mathbf{F} = P \mathbf{i} + Q \mathbf{j} + R \mathbf{k} \). The curl of \( \mathbf{F} \), represented as \( abla \times \mathbf{F} \), is computed using a cross product of the del operator \( abla \) with the vector field \( \mathbf{F} \). For \( \mathbf{F} \), the mathematical expression for the curl is:\[ abla \times \mathbf{F} = \left( \frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z} \right) \mathbf{i} + \left( \frac{\partial P}{\partial z} - \frac{\partial R}{\partial x} \right) \mathbf{j} + \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) \mathbf{k} \]This operation yields a new vector that describes the local rotation of \( \mathbf{F} \). It's important to note that having a non-zero curl implies rotational characteristics, which can relate to rotational physical phenomena such as fluid vortices.

Divergence-Free Condition

The divergence-free condition is a crucial concept when analyzing whether a vector field can be characterized by a particular curl. The divergence of a vector field \( \mathbf{C} \) is calculated as \( abla \cdot \mathbf{C} \), which measures how much the field "spreads out" from a given point. When the divergence is zero, the field is said to be divergence-free. In the context of curls, if a vector field \( \mathbf{F} \) has a curl equal to another vector field \( \mathbf{C} \), then \( \mathbf{C} \) must be divergence-free for this equality to hold.

• Mathematically, \( abla \cdot \left( abla \times \mathbf{F} \right) = 0 \) holds for any vector field \( \mathbf{F} \).
• If \( \mathbf{C} \) is not divergence-free, it means it cannot be expressed as the curl of any vector field.

In our exercise, since \( \mathbf{C} = x \mathbf{i} + y \mathbf{j} + z \mathbf{k} \) has a divergence of 3 (not zero), it proves that such a \( \mathbf{C} \) cannot be a curl of any vector field.

Twice-Differentiable Components

Twice-differentiable functions are those that you can differentiate twice, and each time, the result is still continuous. In vector calculus, twice-differentiable functions become crucial when dealing with conditions like Stokes' Theorem or in determining if a vector field can be expressed as a gradient or a curl. For a vector field \( \mathbf{F} = P \mathbf{i} + Q \mathbf{j} + R \mathbf{k} \) where each component \( P, Q, \) and \( R \) is a function of \( x, y, \) and \( z \), the requirement that these components be twice-differentiable ensures certain properties:

• Ensures the smoothness of the field which is crucial when calculating derivatives like the curl.
• It means both \( \frac{\partial^2}{\partial x^2}, \frac{\partial^2}{\partial y^2} \) and \( \frac{\partial^2}{\partial z^2} \) of the components exist and are well-behaved.

These conditions ensure that the curl and other differential operations yield meaningful results. In the given exercise, irrespective of differentiability, the divergence-free condition not being met was sufficient to conclude the non-existence of such a vector field for the given curl.