Answer :
To solve for the divergence of the vector field [tex]\(\frac{\bar{a} \times \bar{r}}{r^n}\)[/tex], we start by understanding the components involved.
Given:
- [tex]\(\bar{r} = x \mathbf{i} + y \mathbf{j} + z \mathbf{k}\)[/tex]
- [tex]\(r = |\bar{r}| = \sqrt{x^2 + y^2 + z^2}\)[/tex]
- [tex]\(\bar{a} = a_x \mathbf{i} + a_y \mathbf{j} + a_z \mathbf{k}\)[/tex]
- We need to find [tex]\(\operatorname{div}\left(\frac{\bar{a} \times \bar{r}}{r^n}\right)\)[/tex]
The cross product [tex]\(\bar{a} \times \bar{r}\)[/tex] is given by:
[tex]\[ \bar{a} \times \bar{r} = \left|\begin{array}{ccc} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ a_x & a_y & a_z \\ x & y & z \\ \end{array}\right| = (a_y z - a_z y)\mathbf{i} + (a_z x - a_x z)\mathbf{j} + (a_x y - a_y x)\mathbf{k} \][/tex]
This can be written as:
[tex]\[ \bar{a} \times \bar{r} = [(a_y z - a_z y), (a_z x - a_x z), (a_x y - a_y x)] \][/tex]
Now, consider the vector field:
[tex]\[ \mathbf{F} = \frac{\bar{a} \times \bar{r}}{r^n} = \left( \frac{a_y z - a_z y}{r^n}, \frac{a_z x - a_x z}{r^n}, \frac{a_x y - a_y x}{r^n} \right) \][/tex]
We need to compute [tex]\(\operatorname{div}(\mathbf{F})\)[/tex], which is:
[tex]\[ \operatorname{div}(\mathbf{F}) = \frac{\partial}{\partial x} \left( \frac{a_y z - a_z y}{r^n} \right) + \frac{\partial}{\partial y} \left( \frac{a_z x - a_x z}{r^n} \right) + \frac{\partial}{\partial z} \left( \frac{a_x y - a_y x}{r^n} \right) \][/tex]
To compute these partial derivatives, use the quotient rule:
[tex]\[ \frac{\partial}{\partial u} \left( \frac{g(v)}{h(v)} \right) = \frac{h(v) \frac{\partial g(v)}{\partial u} - g(v) \frac{\partial h(v)}{\partial u}}{h(v)^2} \][/tex]
Let's calculate each term separately:
1. [tex]\(\frac{\partial}{\partial x} \left( \frac{a_y z - a_z y}{r^n} \right)\)[/tex]:
- [tex]\(g(x, y, z) = a_y z - a_z y\)[/tex]
- [tex]\(h(x, y, z) = r^n\)[/tex]
[tex]\[ \frac{\partial}{\partial x} (a_y z - a_z y) = 0 \][/tex]
[tex]\[ \frac{\partial}{\partial x} r^n = n r^{n-2} x \][/tex]
[tex]\[ \frac{\partial}{\partial x} \left( \frac{a_y z - a_z y}{r^n} \right) = \frac{(r^n \cdot 0 - (a_y z - a_z y) \cdot n r^{n-2} x)}{r^{2n}} = -\frac{n x (a_y z - a_z y)}{r^{n+2}} \][/tex]
Similarly, for other terms:
2. [tex]\(\frac{\partial}{\partial y} \left( \frac{a_z x - a_x z}{r^n} \right) = -\frac{n y (a_z x - a_x z)}{r^{n+2}}\)[/tex]
3. [tex]\(\frac{\partial}{\partial z} \left( \frac{a_x y - a_y x}{r^n} \right) = -\frac{n z (a_x y - a_y x)}{r^{n+2}}\)[/tex]
Adding these together:
[tex]\[ \operatorname{div}(\mathbf{F}) = -\frac{n x (a_y z - a_z y)}{r^{n+2}} - \frac{n y (a_z x - a_x z)}{r^{n+2}} - \frac{n z (a_x y - a_y x)}{r^{n+2}} \][/tex]
Notice that each term involves a specific arrangement of constants and variables, leading to the cancellation due to the linearity of the vector components and symmetry. Upon simplifying:
[tex]\[ \operatorname{div}(\mathbf{F}) = 0 \][/tex]
Thus, the divergence of the given vector field is:
[tex]\[ \operatorname{div} \left( \frac{\bar{a} \times \bar{r}}{r^n} \right) = 0 \][/tex]
Given:
- [tex]\(\bar{r} = x \mathbf{i} + y \mathbf{j} + z \mathbf{k}\)[/tex]
- [tex]\(r = |\bar{r}| = \sqrt{x^2 + y^2 + z^2}\)[/tex]
- [tex]\(\bar{a} = a_x \mathbf{i} + a_y \mathbf{j} + a_z \mathbf{k}\)[/tex]
- We need to find [tex]\(\operatorname{div}\left(\frac{\bar{a} \times \bar{r}}{r^n}\right)\)[/tex]
The cross product [tex]\(\bar{a} \times \bar{r}\)[/tex] is given by:
[tex]\[ \bar{a} \times \bar{r} = \left|\begin{array}{ccc} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ a_x & a_y & a_z \\ x & y & z \\ \end{array}\right| = (a_y z - a_z y)\mathbf{i} + (a_z x - a_x z)\mathbf{j} + (a_x y - a_y x)\mathbf{k} \][/tex]
This can be written as:
[tex]\[ \bar{a} \times \bar{r} = [(a_y z - a_z y), (a_z x - a_x z), (a_x y - a_y x)] \][/tex]
Now, consider the vector field:
[tex]\[ \mathbf{F} = \frac{\bar{a} \times \bar{r}}{r^n} = \left( \frac{a_y z - a_z y}{r^n}, \frac{a_z x - a_x z}{r^n}, \frac{a_x y - a_y x}{r^n} \right) \][/tex]
We need to compute [tex]\(\operatorname{div}(\mathbf{F})\)[/tex], which is:
[tex]\[ \operatorname{div}(\mathbf{F}) = \frac{\partial}{\partial x} \left( \frac{a_y z - a_z y}{r^n} \right) + \frac{\partial}{\partial y} \left( \frac{a_z x - a_x z}{r^n} \right) + \frac{\partial}{\partial z} \left( \frac{a_x y - a_y x}{r^n} \right) \][/tex]
To compute these partial derivatives, use the quotient rule:
[tex]\[ \frac{\partial}{\partial u} \left( \frac{g(v)}{h(v)} \right) = \frac{h(v) \frac{\partial g(v)}{\partial u} - g(v) \frac{\partial h(v)}{\partial u}}{h(v)^2} \][/tex]
Let's calculate each term separately:
1. [tex]\(\frac{\partial}{\partial x} \left( \frac{a_y z - a_z y}{r^n} \right)\)[/tex]:
- [tex]\(g(x, y, z) = a_y z - a_z y\)[/tex]
- [tex]\(h(x, y, z) = r^n\)[/tex]
[tex]\[ \frac{\partial}{\partial x} (a_y z - a_z y) = 0 \][/tex]
[tex]\[ \frac{\partial}{\partial x} r^n = n r^{n-2} x \][/tex]
[tex]\[ \frac{\partial}{\partial x} \left( \frac{a_y z - a_z y}{r^n} \right) = \frac{(r^n \cdot 0 - (a_y z - a_z y) \cdot n r^{n-2} x)}{r^{2n}} = -\frac{n x (a_y z - a_z y)}{r^{n+2}} \][/tex]
Similarly, for other terms:
2. [tex]\(\frac{\partial}{\partial y} \left( \frac{a_z x - a_x z}{r^n} \right) = -\frac{n y (a_z x - a_x z)}{r^{n+2}}\)[/tex]
3. [tex]\(\frac{\partial}{\partial z} \left( \frac{a_x y - a_y x}{r^n} \right) = -\frac{n z (a_x y - a_y x)}{r^{n+2}}\)[/tex]
Adding these together:
[tex]\[ \operatorname{div}(\mathbf{F}) = -\frac{n x (a_y z - a_z y)}{r^{n+2}} - \frac{n y (a_z x - a_x z)}{r^{n+2}} - \frac{n z (a_x y - a_y x)}{r^{n+2}} \][/tex]
Notice that each term involves a specific arrangement of constants and variables, leading to the cancellation due to the linearity of the vector components and symmetry. Upon simplifying:
[tex]\[ \operatorname{div}(\mathbf{F}) = 0 \][/tex]
Thus, the divergence of the given vector field is:
[tex]\[ \operatorname{div} \left( \frac{\bar{a} \times \bar{r}}{r^n} \right) = 0 \][/tex]
