91Ó°ÊÓ

In multivariable calculus, the gradient vector plays a key role, especially when discussing surfaces and curves. The gradient of a function \( f(x, y, z) \) is denoted as \( abla f \) and is a vector that represents the direction of the steepest ascent of the function. It is composed of the partial derivatives of the function:

• \( \frac{\partial f}{\partial x} \)
• \( \frac{\partial f}{\partial y} \)
• \( \frac{\partial f}{\partial z} \)

In our example, the function \( f(x, y, z) = x^2 + y^2 - z \) leads to the gradient \( abla f = (2x, 2y, -1) \). This vector gives us a crucial insight: it is perpendicular to the surface defined by \( f(x, y, z) = 3 \). This perpendicularity is what helps in finding the normal directions on the surface. By examining the gradient vector, you can determine how the function changes most rapidly and understand the normal directions to a surface.

The velocity vector describes how a position vector changes with respect to time. It is found by differentiating the position vector of a curve with respect to time, \( t \). This results in a vector that tells you the direction and speed at which a point moves along the curve. For the curve \( \mathbf{r}(t) = \sqrt{t} \mathbf{i} + \sqrt{t} \mathbf{j} - \frac{1}{4}(t+3) \mathbf{k} \), the velocity vector is derived as follows:

• \( \frac{d}{dt}(\sqrt{t}) = \frac{1}{2\sqrt{t}} \)
• \( \frac{d}{dt}\left(-\frac{1}{4}(t+3)\right) = -\frac{1}{4} \)

Hence, the velocity vector is \( \mathbf{v}(t) = \frac{1}{2\sqrt{t}} \mathbf{i} + \frac{1}{2\sqrt{t}} \mathbf{j} - \frac{1}{4} \mathbf{k} \). At \( t = 1 \), the velocity vector is \( \mathbf{v}(1) = \frac{1}{2} \mathbf{i} + \frac{1}{2} \mathbf{j} - \frac{1}{4} \mathbf{k} \). This vector gives us a snapshot of the motion along the curve at \( t = 1 \).

Understanding scalar multiples is crucial when comparing two vectors. A vector \( \mathbf{v} \) is a scalar multiple of another vector \( \mathbf{w} \) if there exists a scalar \( \lambda \) such that: \[ \mathbf{v} = \lambda \mathbf{w} \]In this exercise, our goal is to determine if the velocity vector \( \mathbf{v}(1) \) is a scalar multiple of the gradient vector \( abla f(1) \). Both vectors must point in the same or directly opposite direction, differing only by the factor \( \lambda \). Solving the equations \( \lambda \times 2 = \frac{1}{2} \), \( \lambda \times 2 = \frac{1}{2} \), and \( \lambda \times (-1) = -\frac{1}{4} \) yields \( \lambda = \frac{1}{4} \), confirming that they are scalar multiples, indicating that the curve is normal to the surface at the point of intersection.

Partial derivatives are derivatives of functions with more than one variable, with respect to one of those variables, treating the others as constants. They form the building blocks of gradient vectors in multivariable calculus. In the function \( f(x, y, z) = x^2 + y^2 - z \), we calculate:

• \( \frac{\partial f}{\partial x} = 2x \)
• \( \frac{\partial f}{\partial y} = 2y \)
• \( \frac{\partial f}{\partial z} = -1 \)

These derivatives suggest how the function \( f \) changes as we vary one variable while keeping the others constant. Partial derivatives provide us with directional rates of change, which are crucial for constructing the gradient vector that helps in understanding the behavior of functions over surfaces.