Let's analyze the mathematical problem shown in the image step-by-step.

(i) Analyzing the vector field and determining its properties:

The force field $\vec{F}(\vec{r})$ is given by: $\vec{F}(\vec{r}) = -\frac{GmM}{r^3} \vec{r}$ where:

• $G$ is the gravitational constant,
• $r = \left( \sum x_i^2 \right)^{0.5}$ is the magnitude of the position vector,
• $\vec{r}$ is the position vector at $(5, 2, 1)$.

To analyze the curl, the problem is asking you to check the curl of the force field, $\nabla \times \vec{F}(\vec{r})$. For a conservative force field, the curl $\nabla \times \vec{F}$ should equal zero. Since gravitational forces are central forces, they are typically conservative, which means $\nabla \times \vec{F} = 0$. This makes the force field conservative, not solenoidal or ellipsoidal.

Answer: The force field is Conservative.

(ii) Evaluating $\nabla \cdot \vec{r}$:

Next, you are asked to compute $\nabla \cdot \vec{r}$, the divergence of the position vector $\vec{r}$.

The divergence of a position vector $\vec{r} = (x, y, z)$ is: $\nabla \cdot \vec{r} = \frac{\partial x}{\partial x} + \frac{\partial y}{\partial y} + \frac{\partial z}{\partial z} = 3$ So the answer is 3.0000 to four significant figures.

(iii) Evaluating $\nabla \cdot (\vec{r}/r^4)$:

Now, let's calculate $\nabla \cdot \left( \frac{\vec{r}}{r^4} \right)$.

We use the identity for the divergence of a vector field of the form $\frac{\vec{r}}{r^n}$: $\nabla \cdot \left( \frac{\vec{r}}{r^n} \right) = \frac{3 - n}{r^n}$ Here $n = 4$, so: $\nabla \cdot \left( \frac{\vec{r}}{r^4} \right) = \frac{3 - 4}{r^4} = -\frac{1}{r^4}$ Given that $r = \sqrt{5^2 + 2^2 + 1^2} = \sqrt{30} \approx 5.4772$, we substitute this value of $r$: $\nabla \cdot \left( \frac{\vec{r}}{r^4} \right) = -\frac{1}{(5.4772)^4} \approx -0.00110$ Thus, the answer is -0.00110 to four significant figures.

Do you have any questions or would you like further details?

Relative Questions:

1. How do we calculate the curl of a vector field in 3D space?
2. Why are gravitational forces considered conservative?
3. What is the physical meaning of divergence in vector fields?
4. How do you interpret the result of a curl being zero in a vector field?
5. What are the general applications of the vector calculus identities used in this problem?

Tip:

Always verify the form of vector identities before using them in calculations, especially when dealing with non-constant magnitude vectors like $\frac{\vec{r}}{r^n}$.