Answer
The differential form is exact and therefore integrable.
Solution
To prove the integrability of the given differential form
\[
\omega = \frac{y z}{x^{2}+y^{2}} d x - \frac{x z}{x^{2}+y^{2}} d y - \tan^{-1} \frac{y}{x} d z = 0,
\]
we will check if it is an exact differential form. A differential form is exact if it can be expressed as the differential of some function \( f(x, y, z) \).
### Step 1: Identify the components
The differential form can be expressed as:
\[
\omega = P \, dx + Q \, dy + R \, dz,
\]
where:
- \( P = \frac{y z}{x^{2}+y^{2}} \)
- \( Q = -\frac{x z}{x^{2}+y^{2}} \)
- \( R = -\tan^{-1} \frac{y}{x} \)
### Step 2: Check for exactness
For the form to be exact, the following conditions must hold:
1. \( \frac{\partial P}{\partial y} = \frac{\partial Q}{\partial x} \)
2. \( \frac{\partial P}{\partial z} = \frac{\partial R}{\partial x} \)
3. \( \frac{\partial Q}{\partial z} = \frac{\partial R}{\partial y} \)
Let's calculate these partial derivatives.
### Step 3: Calculate the partial derivatives
1. **Calculate \( \frac{\partial P}{\partial y} \)**:
\[
P = \frac{y z}{x^{2}+y^{2}} \implies \frac{\partial P}{\partial y} = \frac{z(x^{2}+y^{2}) - y z(2y)}{(x^{2}+y^{2})^{2}} = \frac{z(x^{2} - y^{2})}{(x^{2}+y^{2})^{2}}
\]
2. **Calculate \( \frac{\partial Q}{\partial x} \)**:
\[
Q = -\frac{x z}{x^{2}+y^{2}} \implies \frac{\partial Q}{\partial x} = -\frac{z(x^{2}+y^{2}) - xz(2x)}{(x^{2}+y^{2})^{2}} = -\frac{z(y^{2} - x^{2})}{(x^{2}+y^{2})^{2}}
\]
3. **Check if \( \frac{\partial P}{\partial y} = \frac{\partial Q}{\partial x} \)**:
\[
\frac{\partial P}{\partial y} = \frac{z(x^{2} - y^{2})}{(x^{2}+y^{2})^{2}}, \quad \frac{\partial Q}{\partial x} = -\frac{z(y^{2} - x^{2})}{(x^{2}+y^{2})^{2}}
\]
These are equal, as \( \frac{\partial P}{\partial y} = -\frac{\partial Q}{\partial x} \).
4. **Calculate \( \frac{\partial P}{\partial z} \)**:
\[
\frac{\partial P}{\partial z} = \frac{y}{x^{2}+y^{2}}
\]
5. **Calculate \( \frac{\partial R}{\partial x} \)**:
\[
R = -\tan^{-1} \frac{y}{x} \implies \frac{\partial R}{\partial x} = -\frac{-y}{x^{2}+y^{2}} = \frac{y}{x^{2}+y^{2}}
\]
6. **Check if \( \frac{\partial P}{\partial z} = \frac{\partial R}{\partial x} \)**:
Both are equal.
7. **Calculate \( \frac{\partial Q}{\partial z} \)**:
\[
\frac{\partial Q}{\partial z} = -\frac{x}{x^{2}+y^{2}}
\]
8. **Calculate \( \frac{\partial R}{\partial y} \)**:
\[
\frac{\partial R}{\partial y} = -\frac{x}{x^{2}+y^{2}}
\]
9. **Check if \( \frac{\partial Q}{\partial z} = \frac{\partial R}{\partial y} \)**:
Both are equal.
### Conclusion
Since all the conditions for exactness are satisfied, the differential form
\[
\frac{y z}{x^{2}+y^{2}} d x - \frac{x z}{x^{2}+y^{2}} d y - \tan^{-1} \frac{y}{x} d z = 0
\]
is exact, which implies that it is integrable. Thus, we have proven the integrability of the given differential form.
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