Problem 21 - Calculus: Early Transcendental Functions

Question

Determine whether the given vector field is conservative and/or incompressible. $$\left(x y^{2}, 3 x z, 4-z y^{2}\right)$$

Step-by-Step Solution

Answer

The given vector field \(\left(x y^{2}, 3 x z, 4-z y^{2}\right)\) is neither conservative nor incompressible.

Step 1: Calculate the Curl of the Vector Field

The curl of a vector field \(F(x, y, z) = (P, Q, R)\) is given by: \[\nabla \times F = \left(\frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z}\right)\mathbf{i} - \left(\frac{\partial P}{\partial z} - \frac{\partial R}{\partial x}\right)\mathbf{j} + \left(\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\right)\mathbf{k}\].

Step 2: Calculate the curl for the given vector field

Substitute \(P = x y^2, Q = 3 x z, R = 4 - z y^2\) into the curl formula to get: \[\nabla \times F = \left(0 - 3x\right)\mathbf{i} - \left(0 - (-y^2)\right)\mathbf{j} + \left(3z - 2xy\right)\mathbf{k} = -3x\mathbf{i} + y^2\mathbf{j} + (3z - 2xy)\mathbf{k}\].

Step 3: Determine if the Vector Field is Conservative

A vector field is conservative if the curl is equal to zero. For the given vector field, the curl is \(-3x\mathbf{i} + y^2\mathbf{j} + (3z - 2xy)\mathbf{k}\), which is not zero, so the vector field is not conservative.

Step 4: Calculate the Divergence of the Vector Field

The divergence of a vector field \(F(x, y, z) = (P, Q, R)\) is given by: \[\nabla \cdot F = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} + \frac{\partial R}{\partial z}\].

Step 5: Calculate the divergence for the given vector field

Substitute \(P = x y^2, Q = 3 x z, R = 4 - z y^2\) into the divergence formula, \[\nabla \cdot F = y^2 + 0 + 0 = y^2\].

Step 6: Determine if the Vector Field is Incompressible

A vector field is incompressible if the divergence is equal to zero. For the given vector field, the divergence is \(y^2\), which is not zero for all values of \(y\). Hence, the vector field is not incompressible.

Key Concepts

Conservative Vector Fields

A vector field is categorized as conservative when the curl of the field is zero. This implies that the field is path-independent, meaning that the work done moving along a path in the vector field depends only on the starting and ending points, not the path taken. In mathematical terms, a conservative vector field can be given as the gradient of a scalar potential function, denoted as:

• If \( abla \times F = 0 \), then the field is conservative.

For the vector field \( (x y^2, 3 x z, 4 - z y^2) \), when we calculated the curl, it resulted in \(-3x\mathbf{i} + y^2\mathbf{j} + (3z - 2xy)\mathbf{k}\), which is not zero. Thus, this vector field is not conservative. Knowing whether a field is conservative is essential for solving problems involving potential energy, electric fields, and fluid flow, where energy conservation principles apply.

Curl of a Vector Field

The curl of a vector field measures the rotation or the swirling strength of the field at a point. It is a vector that describes the infinitesimal rotation around that point. The formula for the curl \( abla \times F \) of a vector field \( F(x, y, z) = (P, Q, R) \) is given by:

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

When we calculated the curl for the given vector field \( (x y^2, 3 x z, 4 - z y^2) \), we obtained \(-3x\mathbf{i} + y^2\mathbf{j} + (3z - 2xy)\mathbf{k}\). This non-zero curl indicates that the field has a rotational component, meaning the field "turns" around some axis. Curl is extensively used in fluid mechanics to determine the rotation of fluid elements.

Divergence of a Vector Field

Divergence is a scalar measure of a vector field's tendency to "spread out" from or "converge" into points in space. It specifically measures the magnitude of a source or sink at a given point. In mathematical notation, the divergence \( abla \cdot F \) of a vector field \( F(x, y, z) = (P, Q, R) \) is:

• \( abla \cdot F = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} + \frac{\partial R}{\partial z} \)

For the vector field \( (x y^2, 3 x z, 4 - z y^2) \), the divergence calculated was \( y^2 \), which varies depending on the value of \( y \). Since the divergence is not always zero, the vector field suggests the presence of sources or sinks. Divergence is used in many applications, such as determining the behavior of electric and magnetic fields in electromagnetism and analyzing fluid flow.

Incompressible Vector Fields

An incompressible vector field is characterized by a zero divergence. This means that the field neither compresses nor expands at any point, maintaining a constant density. Such fields are also often referred to as solenoidal fields. The conditions for an incompressible vector field are:

• If \( abla \cdot F = 0 \), then the field is incompressible.

In the analysis of our vector field \( (x y^2, 3 x z, 4 - z y^2) \), the divergence was found to be \( y^2 \), clearly indicating it is not zero for all \( y \). Therefore, this vector field is not incompressible because it exhibits variance in density with changes in \( y \). Recognizing incompressible fields is vital in applications involving fluid dynamics, where incompressible flow assumptions lead to simplified and fundamental solutions to flow problems.