Question:
If \( u = \tan^{-1}\left(\frac{y}{x}\right) \), then \( x \frac{\partial u}{\partial x} + y \frac{\partial u}{\partial y} = \text{?} \)
यदि \( u = \tan^{-1}\left(\frac{y}{x}\right) \) है, तो \( x \frac{\partial u}{\partial x} + y \frac{\partial u}{\partial y} = \text{?} \)
Options:
- Sin 2u
- 0
- None of the given options
- Cos 2u
- Sin 2u
- 0
- दिए गए विकल्पों में से कोई नहीं
- Cos 2u
Correct Answer: b)
Explanation:
The function \( u(x, y) = \tan^{-1}\left(\frac{y}{x}\right) \) is homogeneous of degree \( n = 0 \) since \( u(tx, ty) = \tan^{-1}\left(\frac{ty}{tx}\right) = \tan^{-1}\left(\frac{y}{x}\right) = t^0 u(x, y) \). By Euler’s Theorem for homogeneous functions, \[ x \frac{\partial u}{\partial x} + y \frac{\partial u}{\partial y} = n \cdot u = 0 \cdot u = 0 \]
फलन \( u(x, y) = \tan^{-1}\left(\frac{y}{x}\right) \) शून्य घात (\( n = 0 \)) का एक समघात फलन (homogeneous function) है क्योंकि \( u(tx, ty) = t^0 u(x, y) \)। समघात फलनों के लिए यूलर के प्रमेय के अनुसार: \[ x \frac{\partial u}{\partial x} + y \frac{\partial u}{\partial y} = n \cdot u = 0 \cdot u = 0 \]