The problem you have shared involves finding the value of $x \frac{\partial u}{\partial x} + y \frac{\partial u}{\partial y} + z \frac{\partial u}{\partial z}$ where $u = \sin^{-1} \left( \frac{x + 2y + 3z}{x^2 + y^2 + z^2} \right)$.

This expression resembles Euler's theorem for homogeneous functions, so it suggests the function's degree of homogeneity.

Step-by-step approach:

1. Expression for $u$: $u = \sin^{-1} \left( \frac{x + 2y + 3z}{x^2 + y^2 + z^2} \right)$ Let’s denote the argument inside the inverse sine as: $f(x, y, z) = \frac{x + 2y + 3z}{x^2 + y^2 + z^2}$

2. Find Partial Derivatives: We need to compute $\frac{\partial u}{\partial x}, \frac{\partial u}{\partial y}, \frac{\partial u}{\partial z}$, which can be found using the chain rule and implicit differentiation since $u = \sin^{-1}(f)$.

3. Apply the formula: We are asked to evaluate: $x \frac{\partial u}{\partial x} + y \frac{\partial u}{\partial y} + z \frac{\partial u}{\partial z}$ Using Euler's theorem on homogeneous functions, it suggests a simplification related to the degree of the function inside the sine inverse.

I'll compute this step-by-step.The value of the expression $x \frac{\partial u}{\partial x} + y \frac{\partial u}{\partial y} + z \frac{\partial u}{\partial z}$ simplifies to:

$-\frac{x + 2y + 3z}{\sqrt{\frac{x^4 + 2x^2y^2 + 2x^2z^2 - x^2 - 4xy - 6xz + y^4 + 2y^2z^2 - 4y^2 - 12yz + z^4 - 9z^2}{x^4 + 2x^2y^2 + 2x^2z^2 + y^4 + 2y^2z^2 + z^4}} \cdot (x^2 + y^2 + z^2)}$

This is quite a complex expression, but the key insight is that it evaluates the directional derivative of the function $u$ along the vector $(x, y, z)$. Further simplification could involve specific values of $x, y, z$, depending on your context.

Would you like more details on how the partial derivatives were computed, or on any other step?

Here are 5 related questions that could help expand your understanding:

1. What is the geometric interpretation of this expression in terms of gradients and directional derivatives?
2. How would the result change if the function $u$ were homogeneous of a different degree?
3. Can this expression be simplified for specific values of $x, y, z$?
4. How would the result differ if a different function of $x, y, z$ were used instead of $u$?
5. How do we apply Euler's theorem to homogeneous functions in this context?

Tip: When differentiating complex multivariable functions, applying the chain rule carefully can lead to important simplifications based on the problem's symmetry.