91Ó°ÊÓ

The gradient vector plays a crucial role in understanding how functions change. In the context of a scalar field like ours, the gradient vector helps identify the direction of steepest ascent. It's derived using partial derivatives of the function with respect to each variable. For the function given, \( F(x, y, z) = z - x^2 - y^2 \), the gradient vector is calculated as follows:

• The partial derivative with respect to \( x \) is \( \frac{\partial F}{\partial x} = -2x \).
• The partial derivative with respect to \( y \) is \( \frac{\partial F}{\partial y} = -2y \).
• The partial derivative with respect to \( z \) is \( \frac{\partial F}{\partial z} = 1 \).

Hence, the gradient vector is \( abla F = (-2x, -2y, 1) \). It gives us valuable insight by pointing perpendicular to level surfaces. For the specific point \( P(2, -2, 1) \), substituting its coordinates yields \( abla F|_P = (-4, 4, 1) \). This indicates the normal direction to the paraboloid at that point.

A paraboloid is a three-dimensional surface representing quadratic surfaces. In this exercise, the equation \( z = x^2 + y^2 - 7 \) describes an upward-opening paraboloid. This surface forms by revolving a parabola around its axis. The term \( z = x^2 + y^2 - 7 \) shows that the vertex of the paraboloid is at the point \( (0, 0, -7) \). This is where it changes direction from dipping downward to rising upward.

• The shape is symmetrical around the z-axis.
• The further from the vertex, the higher the z-value.

Visualizing this structure can help understand how it interacts with any point on the surface, including how the gradient vector connects with and indicates the normal to this curved surface.

A scalar field assigns a single value (a scalar) to every point in a space. It contrasts with vector fields, which assign a vector to each point. In this problem, our scalar field is given by \( F(x, y, z) = z - x^2 - y^2 \). Think of a scalar field like a temperature map, where every location has a specific temperature.

• The value of the function at a point represents a level of the scalar field.
• The scalar field helps determine the "elevation" at various coordinates.

In particular, for point \( P(2, -2, 1) \), the scalar field value of \( F \) is \(-7\). This reveals where on the surface this point lies (level surface) and impacts the orientation of the gradient vector.

Partial derivatives are essential to calculate the gradient vector. They measure how a function changes as each individual variable changes, while holding others constant. For the scalar field \( F(x, y, z) = z - x^2 - y^2 \), partial derivatives are taken as follows:

• \( \frac{\partial F}{\partial x} = -2x \) tells us how \( F \) changes with \( x \), treating \( y \) and \( z \) as constants.
• \( \frac{\partial F}{\partial y} = -2y \) shows how \( F \) changes with \( y \).
• \( \frac{\partial F}{\partial z} = 1 \) indicates how \( F \) changes with \( z \).

These derivatives inform us about the function's slope along each axis. By combining them, you get the gradient vector, indicating the steepest ascent direction on the surface.