Problem 22 - Calculus: Early Transcendental Functions

Question

Determine whether the given vector field is conservative and/or incompressible. $$\langle x, y, 1-3 z\rangle$$

Step-by-Step Solution

Answer

The given vector field \(\langle x, y, 1-3 z\rangle\) is conservative but not incompressible.

Step 1: Identify and write down the vector field

Here, the vector field \(F\) given in the exercise is \(\langle x, y, 1-3 z\rangle\).

Step 2: Calculate the curl of the vector field

The curl of a vector field \(F = \langle P, Q, R \rangle\) is given by \(del cross F = \langle (R_y - Q_z), (P_z - R_x), (Q_x - P_y) \rangle\). Using the given vector field \(F = \langle x, y, 1-3 z\rangle\), we can calculate its curl as \(del cross F = \langle 0, 0, 0 \rangle\).

Step 3: Determine if the vector field is conservative

A vector field is considered conservative if its curl, \( del cross F\) equals to the zero vector. Here, since \( del cross F = \langle 0, 0, 0 \rangle\), the given vector field \(F\) is thus conservative.

Step 4: Calculate the Divergence of the vector field

Divergence of a vector field \(F = \langle P, Q, R \rangle\) is given by \(del dot F = P_x + Q_y + R_z\). Using the given vector field parameters, we can calculate the divergence of the vector field as \(del dot F = 1 + 1 + (-3) = -1\).

Step 5: Determine if the vector field is incompressible

A vector field is incompressible if its divergence, \(del dot F\) equals to zero. Here, as \(del dot F = -1\), the given vector field \(F\) is not incompressible.

Key Concepts

Conservative Vector Field

A vector field is known as conservative if it can be expressed as the gradient of a scalar potential function. In simpler terms, this means there exists a function whose gradient matches the vector field. This property is closely related to the physical concept of energy conservation in forces.

• One essential characteristic of a conservative vector field is its curl. If the curl of the vector field is a zero vector, the field is deemed conservative.
• For instance, with our vector field \(\langle x, y, 1-3z\rangle\), its curl was calculated as \(\langle 0, 0, 0 \rangle\).
• Consequently, since its curl is equal to zero, the vector field is conservative.

Conservative vector fields have interesting properties such as path independence. This means that the line integral between two points remains constant, regardless of the path taken.

Incompressible Vector Field

An incompressible vector field is one where the divergence is zero. In practical terms, it implies that the field has no net volume flow—imagine an ideal fluid with constant density. The divergence measures the rate of expansion or contraction at any point.

• For a vector field given by \(\langle P, Q, R \rangle\), the divergence is calculated as \(P_x + Q_y + R_z\).
• Using our example, the vector field \(\langle x, y, 1-3z\rangle\) has a divergence of \(-1\) as calculated: \(1 + 1 - 3 = -1\).
• Because the divergence is not zero, this vector field is not incompressible.

Understanding incompressibility is vital in fields like fluid dynamics, ensuring the conservation of mass in ideal fluid flows.

Curl of a Vector Field

The curl of a vector field provides a measure of the field's rotational tendency. Imagine placing a small paddle wheel in the field. If the wheel spins, there's a rotational effect or 'curl'.

• Mathematically, with a vector field \(\langle P, Q, R \rangle\), the curl can be found using the expression: \(\langle R_y - Q_z, P_z - R_x, Q_x - P_y \rangle\).
• For our vector field \(\langle x, y, 1-3z \rangle\), the curl turns out to be \(\langle 0, 0, 0 \rangle\).
• This implies that the field has no rotational component.

If a vector field has a zero curl, it indicates a potential for path independence and is a key criterion for a conservative field.

Divergence of a Vector Field

The divergence of a vector field quantifies how much the field is spreading out from a point. Consider it like measuring the flow out of an imaginary surface surrounding the point.

• For the vector field \(\langle P, Q, R \rangle\), the divergence is calculated as \(P_x + Q_y + R_z\).
• Using the field \(\langle x, y, 1-3z \rangle\), the divergence is \(-1\), calculated as follows: \(1 + 1 - 3 = -1\).
• The negative value suggests more of a net inflow towards the point rather than outflow.

Divergence is vital in understanding sources and sinks of the field, and it plays a crucial role in equations governing fluid flow and electromagnetism.