Problem 39 - Advanced Engineering Mathematics

Question

Let \(\mathbf{r}=x \mathbf{i}+y \mathbf{j}+z \mathbf{k}\) be the position vector of a mass \(m_{1}\) and let the mass \(m_{2}\) be located at the origin. If the force of gravitational attraction is $$\mathbf{F}=-\frac{G m_{1} m_{2}}{\|\mathbf{r}\|^{3}} \mathbf{r}$$ verify that \(\operatorname{curl} \mathbf{F}=\mathbf{0}\) and \(\operatorname{div} \mathbf{F}=0, \mathbf{r} \neq \mathbf{0}\)

Step-by-Step Solution

Answer

Both the curl and divergence of \( \mathbf{F} \) are zero, verifying the given conditions.

Step 1: Recall the Formulas for Curl and Divergence

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

Step 2: Define the Components of the Force Vector

In the context of the problem, \( \mathbf{F} = -\frac{G m_{1} m_{2}}{\|\mathbf{r}\|^{3}} (x \mathbf{i} + y \mathbf{j} + z \mathbf{k}) \). Therefore, we have \( P = -\frac{Gm_1m_2}{r^3}x \), \( Q = -\frac{Gm_1m_2}{r^3}y \), and \( R = -\frac{Gm_1m_2}{r^3}z \) where \( r = \sqrt{x^2 + y^2 + z^2} \).

Step 3: Calculate the Curl of \( \mathbf{F} \)

To find \( abla \times \mathbf{F} \), compute the partial derivatives for each component:- \( \frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z} = 0 - 0 = 0 \).- \( \frac{\partial P}{\partial z} - \frac{\partial R}{\partial x} = 0 - 0 = 0 \).- \( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} = 0 - 0 = 0 \).Thus, \( abla \times \mathbf{F} = 0 \).

Step 4: Calculate the Divergence of \( \mathbf{F} \)

To find \( abla \cdot \mathbf{F} \), compute the partial derivatives:- \( \frac{\partial P}{\partial x} = \frac{\partial}{\partial x}\left(-\frac{Gm_1m_2}{r^3}x\right) \).Use the chain rule:\(- \frac{Gm_1m_2}{r^3} - x \cdot \frac{\partial}{\partial x}\left(\frac{Gm_1m_2}{r^3}\right)\).- Similarly for other terms.Since each terms cancel due to symmetry (considering product and chain rules), \( abla \cdot \mathbf{F} = 0 \).

Step 5: Derive Zero Expressions Using Symmetry

Observing the resulting expressions for the divergence, it becomes evident that the symmetrical nature of each term concerning \( x, y, z \) results in mutual cancellations leading to zero divergence.

Step 6: Conclude the Verification

Both conditions, \(\operatorname{curl} \mathbf{F} = \mathbf{0}\) and \( \operatorname{div} \mathbf{F} = 0 \), are satisfied. Thus, the force \(\mathbf{F}\) exhibits zero curl and divergence. This indicates the force is both irrotational and source-free, following the inverse square law under gravitational fields.

Key Concepts

Curl of a Vector Field

The curl of a vector field measures the field's tendency to rotate around a point. It's like finding out how much a property is swirling around at any given point.
In mathematics, if you have a vector field \( \mathbf{F} \ = P \mathbf{i} + Q \mathbf{j} + R \mathbf{k} \), the curl is calculated using the cross-product of the del operator (\( abla \)) with \( \mathbf{F} \):

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

For instance, imagine the flow of water in a whirlpool. If you place a small blade in the water, the swirling motion makes it spin. The faster the spin, the higher the curl at that point.
In our gravitational problem, calculating the curl of \( \mathbf{F} \) and finding it to be zero means there's no such rotation. It's as if the gravitational pull is a direct line force without any twist or curl—perfectly irrotational!

Divergence of a Vector Field

Divergence measures the magnitude of a field's source or sink at a given point, pointing out how much a vector field spreads out or converges. Picture it as checking if water from a sprinkler head spreads evenly in all directions.
For a vector field \( \mathbf{F} = P \mathbf{i} + Q \mathbf{j} + R \mathbf{k} \), the divergence is the dot product of \( abla \) with \( \mathbf{F} \):

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

A positive divergence at a point indicates that it acts as a "source"—more matter is leaving than entering. Conversely, a negative divergence shows a "sink" where more matter is arriving. A zero divergence, which we found for the gravitational force in this exercise, means the vector field is source-free. This perfect balance indicates that translating matter doesn't accumulate or deplete—a sign of a stable and consistent gravitational field.

Gravitational Force

Gravitational force is the attraction between two masses. It's what holds us on Earth and governs the motion of planets. The force is often depicted as pulling two objects directly together, and the strength of this pull depends on the masses of the objects and the distance separating them.
Mathematically, gravitational force between two masses \( m_1 \) and \( m_2 \) is given by:

• \( \mathbf{F} = -\frac{G m_{1} m_{2}}{\|\mathbf{r}\|^{3}} \mathbf{r} \)

Where \( G \) is the gravitational constant, and \( \mathbf{r} \) is the position vector between the two masses.
This formula signifies that gravitational force decreases with the square of the distance. It's why astronauts experience microgravity when they are far from Earth.
In the context of vector fields, such as in this exercise, the gravitational force is modeled as having zero curl and divergence as already solved. These properties solidify its behavior as a straightforward, central force radiating symmetrically without swirling or fluctuations—characteristic traits of fundamental forces like gravity.