Let's analyze each statement to determine whether it's true or false:

1. True: $f_u(a, b) = || \nabla f(a, b) ||$.

   • This statement is incorrect. $f_u(a, b)$ refers to the directional derivative of $f$ in the direction of a unit vector $\vec{u}$. It is computed as $f_u(a, b) = \nabla f(a, b) \cdot \vec{u} = || \nabla f(a, b) || \cos \theta$, where $\theta$ is the angle between $\nabla f(a, b)$ and $\vec{u}$. Therefore, $f_u(a, b)$ is only equal to $|| \nabla f(a, b) ||$ when $\vec{u}$ is in the same direction as $\nabla f(a, b)$, not in all cases.

2. False: The gradient vector $\nabla f(a, b)$ is tangent to the contour of $f$ at $(a, b)$.

   • This statement is false. The gradient vector $\nabla f(a, b)$ is actually normal (perpendicular) to the contour (or level curve) of $f$ at $(a, b)$, not tangent to it.

3. True: $f_u(a, b)$ is parallel to $\vec{u}$.

   • This statement is correct. The directional derivative $f_u(a, b)$ in the direction of $\vec{u}$ represents the rate of change of $f$ in that direction, meaning the vector is aligned with $\vec{u}$ by definition.

4. False: If $f(x, y)$ has $f_x(a, b) = 0$ and $f_y(a, b) = 0$ at the point $(a, b)$, then $f$ is constant everywhere.

   • This statement is incorrect. Having $f_x(a, b) = 0$ and $f_y(a, b) = 0$ means that $(a, b)$ is a critical point, but it does not imply that $f$ is constant everywhere. $f$ could have a local maximum, minimum, or saddle point at $(a, b)$.

5. True: Suppose $f_x(a, b)$ and $f_y(a, b)$ both exist. Then there is always a direction in which the rate of change of $f$ at $(a, b)$ is zero.

   • This statement is true. Since the rate of change in any direction can be represented by the dot product $\nabla f(a, b) \cdot \vec{u}$, there exists a direction perpendicular to $\nabla f(a, b)$ where the directional derivative will be zero.

6. True: If $\vec{u}$ is a unit vector, then $f_u(a, b)$ is a vector.

   • This statement is incorrect. $f_u(a, b)$ is a scalar value representing the rate of change of $f$ in the direction of $\vec{u}$; it is not a vector.

7. True: If $\vec{u}$ is perpendicular to $\nabla f(a, b)$, then $f_u(a, b) = 0$.

   • This statement is correct. If $\vec{u}$ is perpendicular to $\nabla f(a, b)$, then $\cos \theta = 0$ in the formula $f_u(a, b) = || \nabla f(a, b) || \cos \theta$, resulting in $f_u(a, b) = 0$.

8. False: $\nabla f(a, b)$ is a vector in 3-dimensional space.

   • This statement is incorrect. Since $f(a, b)$ is a function of two variables $(a, b)$, the gradient $\nabla f(a, b)$ is a vector in 2-dimensional space (it has components in the $x$- and $y$-directions only).

Would you like further details on any of these statements or have any additional questions?

Here are 5 questions to expand on this content:

1. How is the directional derivative $f_u(a, b)$ calculated for any given function?
2. What is the significance of the gradient vector in understanding the behavior of a function?
3. Can you explain the role of critical points in determining local maxima and minima?
4. What does it mean geometrically when the directional derivative is zero in a certain direction?
5. How would the gradient vector change if the function were extended to three dimensions?

Tip: Remember that the gradient vector $\nabla f$ always points in the direction of the steepest increase of the function at any given point.