Problem 16 - Calculus: Early Transcendental Functions

Question

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

Step-by-Step Solution

Answer

The vector field is neither conservative nor incompressible.

Step 1: Compute the Curl

To calculate the curl, form a 3x3 matrix with the unit vector \(i, j, k\), the derivative operators \(\frac{d}{dx}\), \(\frac{d}{dy}\), \(\frac{d}{dz}\), and the components of the vector field \((y^2, x^2 e^z, cos (xy))\). The determinant of this matrix gives the curl of the vector field. Hence, \[\text{Curl} = (\frac{dR}{dy} - \frac{dQ}{dz})i - (\frac{dP}{dz} - \frac{dR}{dx})j + (\frac{dQ}{dx} - \frac{dP}{dy})k\].

Step 2: Evaluate the Curl

Substituting values into the curl formula and simplifying, we have \[\text{Curl} = (0 - x^2 e^z)i - (0 - (- \sin(xy)))j + ( 2x e^z - 0)k\]. This simplifies to \[ \text{Curl} = -x^2 e^z i + \sin(xy) j + 2x e^z k\]. Since the curl is not zero, the vector field is not conservative.

Step 3: Compute the Divergence

Divergence of a vector field is given by \(div F = \frac{dP}{dx} + \frac{dQ}{dy} + \frac{dR}{dz}\).

Step 4: Evaluate the Divergence

Substituting values into the divergence formula and simplifying, we have \[div F = 0 + 2y + 0 = 2y.\] Since the divergence is not zero, the vector field is not incompressible.

Key Concepts

Conservative Vector Field

To understand what makes a vector field conservative, imagine it as a road map of forces that have specific properties. A vector field is said to be conservative if it is the gradient of some scalar potential function. This means there is a way to represent the field as the steepness of a surface.
If you can trace from one point to another on this surface and the work done is independent of your path, then you have a conservative vector field. A key property of these fields is that their curl is zero: \\( \text{Curl} \, F = 0 \).

• These fields have no loops; they are irrotational.
• The potential energy varies predictably, like a hill's elevation.
• If given a vector field \((F_1, F_2, F_3)\), it should satisfy \( abla \times \mathbf{F} = 0 \).

In the exercise you have given, because the curl is not zero, the vector field cannot be conservative.

Incompressible Vector Field

Incompressibility in vector fields refers to the concept that a fluid has constant density as it flows through the field. This is often associated with fluid dynamics where the material isn't being squeezed or stretched.
An incompressible vector field is characterized by a divergence of zero: \\( \text{div} \, F = 0 \).

• If you think of fluids, an incompressible field describes a situation where fluid density remains constant.
• The idea can be visualized through liquids like water where volume remains unchanged despite flow.
• For a given field \( \mathbf{F}(x, y, z) = (P, Q, R) \), you find divergence with \( abla \cdot \mathbf{F} = \frac{dP}{dx} + \frac{dQ}{dy} + \frac{dR}{dz} \).

In the provided problem, since the divergence of the field is not zero, it indicates the field is compressible.

Curl and Divergence

Curl and divergence are core operations in vector calculus that help characterize vector fields. Understanding these concepts is vital for manipulating and analyzing vector fields.
The **curl** measures the rotation within the field. If you place a small paddle wheel into a fluid flow represented by the field, the curl tells you if, and how fast, that wheel would spin.

• Think of curl as detecting local spinning or turning; the component of curl perpendicular to the paddle wheel signifies rotation.
• In mathematics, curl is expressed by: \\( abla \times \mathbf{F} \).

On the other hand, the **divergence** measures the rate at which "stuff" exits out of a point in the field. Imagine a balloon in the flow; divergence indicates if the balloon would inflate or deflate.

• Divergence assesses how much a vector field is "spreading out" from a point.
• Mathematically, it's expressed as: \\( abla \cdot \mathbf{F} \).

In your exercise, analyzing these quantities tells us about the behavior and properties of the field: not conservative or incompressible based on their respective operations.