Problem 20 - Calculus: Early Transcendental Functions

Question

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

Step-by-Step Solution

Answer

The given vector field \(\langle 2xz, 3y, x^2 - y\rangle\) is neither conservative nor incompressible.

Step 1: Write down the vector field

The given vector field \(F\) is \(\left\langle 2xz, 3y, x^2 - y \right\rangle\).

Step 2: Check if the vector field is conservative

The Curl of \(F\) is \(\nabla \times F\). Calculate the Curl using the determinant:\[\begin{vmatrix} i & j & k \\frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \2xz & 3y & x^2 - y\end{vmatrix}\]The resulting value is \[(0 - 0)i - (2z-2z)j + (3-0)k = \langle 0, 0, 3\rangle\]The Curl is not equal to zero vector. Thus, the vector field is not conservative.

Step 3: Check if the vector field is incompressible

The Divergence of \(F\) is \(\nabla \cdot F\). This is calculated as \[\frac{\partial}{\partial x}(2xz) + \frac{\partial}{\partial y}(3y) + \frac{\partial}{\partial z}(x^2 - y) = 2z + 3 + 0 = 2z+3\]The Divergence is not zero. Hence, the vector field is not incompressible.

Key Concepts

Conservative Vector Fields

A conservative vector field is a vector field where the work done by moving a particle around a closed path is zero. It also means there exists a scalar potential function, say \( V(x, y, z) \), such that the vector field can be expressed as the gradient of this potential function.
For a vector field \( F \) to be conservative, its curl must equal zero. This characteristic indicates that there are no rotational components in the vector field. In the original problem, the given vector field \( F = \langle 2xz, 3y, x^2 - y \rangle \) was tested for conservativeness. We found that the curl of \( F \) is \( \langle 0, 0, 3 \rangle \), which is not the zero vector.
Therefore, in this case, the vector field is not conservative. Understanding conservative vector fields is particularly important in physics and engineering, as it relates to energy conservation.

Incompressible Vector Fields

An incompressible vector field is one where the divergence of the field is zero at every point. This indicates that the field does not have any sources or sinks; essentially, the field neither spreads out nor concentrates.
Mathematically, a vector field \( F \) is incompressible if \( abla \cdot F = 0 \). In the original solution, after computing the divergence of \( F = \langle 2xz, 3y, x^2 - y \rangle \), the result was \( 2z + 3 \).
Since this is not zero, it signals that the field is not incompressible. Understanding incompressible vector fields is essential in fluid dynamics, where it refers to the flow condition of a fluid.

Curl of a Vector Field

The curl of a vector field is a vector that describes the infinitesimal rotation of the field at a point. Knowing the curl helps us understand how the vector field rotates in space.
The curl is calculated using the determinant of a matrix, which involves partial derivatives of the component functions of the vector field. In the given exercise, this calculation resulted in the vector \( \langle 0, 0, 3 \rangle \).
The non-zero curl suggests that there is rotation in the vector field. The concept of curl is vital for understanding rotational motion in fields like electromagnetism and fluid dynamics.

Divergence of a Vector Field

Divergence measures how much a vector field spreads out from a point. It provides insight into the field's behavior concerning expansion and contraction.
To calculate divergence, you sum up the partial derivatives of each component of the vector field with respect to its respective variable. For the example field \( F = \langle 2xz, 3y, x^2 - y \rangle \), the divergence was found to be \( 2z + 3 \), not zero.
A non-zero divergence indicates that the field has sources or sinks at various points. This is a critical concept in fields like fluid mechanics and electromagnetics, where it helps describe the distribution of physical quantities like fluid flow or electric charge.