Problem 3 - Thomas Calculus

Question

Which fields are conservative, and which are not? \(\mathbf{F}=y \mathbf{i}+(x+z) \mathbf{j}-y \mathbf{k}\)

Step-by-Step Solution

Answer

The field \( \mathbf{F} \) is not conservative because its curl is not zero.

Step 1: Identify if the field is a gradient

A vector field \( \mathbf{F} = P \mathbf{i} + Q \mathbf{j} + R \mathbf{k} \) is conservative if it can be expressed as the gradient of some scalar potential function \( f(x, y, z) \). To determine if a vector field is conservative, we can check if the curl of \( \mathbf{F} \) is zero.

Step 2: Compute the curl of \( \mathbf{F} \)

The curl of a vector field \( \mathbf{F} = y \mathbf{i} + (x+z) \mathbf{j} - y \mathbf{k} \) is given by:\[ abla \times \mathbf{F} = \left( \frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z} \right) \mathbf{i} + \left( \frac{\partial P}{\partial z} - \frac{\partial R}{\partial x} \right) \mathbf{j} + \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) \mathbf{k} \]Substituting the components of \( \mathbf{F} \):\( P = y, \quad Q = x + z, \quad R = -y \)Calculate each component:\[ \frac{\partial R}{\partial y} = -1, \quad \frac{\partial Q}{\partial z} = 1 \]\[ \frac{\partial P}{\partial z} = 0, \quad \frac{\partial R}{\partial x} = 0 \]\[ \frac{\partial Q}{\partial x} = 1, \quad \frac{\partial P}{\partial y} = 1 \]Thus, the curl becomes:\[ abla \times \mathbf{F} = (-1 - 1)\mathbf{i} + (0 - 0)\mathbf{j} + (1 - 1)\mathbf{k} = -2\mathbf{i} \]

Step 3: Conclude about conservativeness

Since the curl of \( \mathbf{F} \) is not zero, specifically \( abla \times \mathbf{F} = -2\mathbf{i} \), the vector field \( \mathbf{F} \) is not conservative, because a conservative field must have a zero curl.

Key Concepts

Curl of a Vector Field

The curl of a vector field gives insights into the rotational tendency of the field at any given point. It's a fundamental concept in vector calculus, especially for determining if a vector field is conservative. The curl of a vector field \( \mathbf{F} = P\mathbf{i} + Q\mathbf{j} + R\mathbf{k} \) is given by:

• \( \left( \frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z} \right) \mathbf{i} \)
• \( \left( \frac{\partial P}{\partial z} - \frac{\partial R}{\partial x} \right) \mathbf{j} \)
• \( \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) \mathbf{k} \)

This calculation essentially measures how much the vector field "twists" around a given point.
If the curl of a vector field results in a non-zero vector, it implies that the field has some rotational characteristics, which can be directly observed in this exercise. For the given vector field, \( abla \times \mathbf{F} = -2\mathbf{i} \), showing that there is rotation along the path described by the field.

Gradient of Scalar Potential

The gradient is an operation that transforms a scalar function into a vector field. It points in the direction of the greatest rate of increase of the function at any point and its magnitude is the slope of the function in that direction. When a vector field can be expressed as the gradient of a scalar potential function, it is called conservative.
For a vector field \( \mathbf{F} \) to be conservative, there must exist a scalar function \( f(x, y, z) \) such that \( \mathbf{F} = abla f \). This implies that the path integral of \( \mathbf{F} \) is path-independent, a key property of conservative fields.
However, this can only be true if the curl of \( \mathbf{F} \) is zero, leading back to our initial finding that the vector field must have no rotational effect for conservativeness. This allows any movement in the field to be describable solely by potential energy without loss or gain due to rotation or external force.

Conditions for Conservativeness

For a vector field to be classified as conservative, certain conditions need to be met, aligning with the field's mathematical properties.

• Zero Curl: As discussed, the curl must be zero \( abla \times \mathbf{F} = 0 \). This indicates the absence of rotation, suggesting that the field can completely be described by a scalar potential function.
• Defined Everywhere: The vector field must be defined and continuous over a simply-connected domain. A simply-connected domain means that any loop within the domain can be continuously contracted to a point without leaving the domain.
• Existence of a Potential Function: There should exist a function \( f(x, y, z) \) such that \( \mathbf{F} = abla f \), allowing energy movement through the field to be calculable and predictable based on potential rather than force applied.

Once these conditions are fulfilled, the field can be deemed conservative and behaves predictably in terms of energy conservation, making calculations and predictions much simpler.