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The gradient of the function \( u = xyz^2 \) at the point \( (1, 0, 3) \) is:

\[
\vec{\nabla} u = (0, 9, 0)
\]

The greatest rate of increase occurs in the direction of the gradient vector, and the magnitude of this vector gives the greatest rate of increase. Since the gradient is \( (0, 9, 0) \), the magnitude is simply:

\[
\text{Magnitude} = \sqrt{0^2 + 9^2 + 0^2} = 9
\]

Thus, the greatest rate of increase is \( \boxed{9} \).

Would you like more details or have any questions?

Here are five related questions:
1. How is the gradient vector related to the direction of maximum increase?
2. What is the physical interpretation of the gradient in 3D space?
3. How do you find the direction of the greatest rate of change for any scalar field?
4. How does the magnitude of the gradient vector change as you move away from the point?
5. How would the gradient change if the function \( u \) had additional terms?

**Tip:** The magnitude of the gradient gives you the steepest ascent, while the direction of the gradient points towards it!

What is greatest rate of increase of u=xyz^2 at the point (1,0,3)?

Find the greatest rate of increase for the function u=xyz^2 at the point (1,0,3). This problem demonstrates how to compute the gradient vector and find its magnitude to determine the steepest ascent in multivariable calculus.

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