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The gradient vector is a critical aspect when working with surfaces and their properties, like tangent planes and normal lines. It is essentially a vector that summarizes how a function changes at any point across its three dimensions. The gradient of a scalar field, such as our surface equation, is obtained by taking the partial derivatives of the function with respect to each variable:

• Partial derivative with respect to x, denoted as \( \frac{\partial f}{\partial x} \).
• Partial derivative with respect to y, denoted as \( \frac{\partial f}{\partial y} \).
• Partial derivative with respect to z, denoted as \( \frac{\partial f}{\partial z} \).

In this problem, our surface function is \( f(x, y, z) = x^2 - xy - y^2 - z \). Calculating these, we find the gradient \( abla f = (2x - y, -x - 2y, -1) \). The gradient essentially points in the direction of the steepest ascent and is perpendicular to the contour line at the point we’re considering.

The normal line to a surface at a given point is a line that is perpendicular to the surface. In the context of a 3D surface, this line can be particularly useful as it provides a direction of maximum change from the surface. The normal vector we calculated from the gradient in this instance serves as the direction vector for our normal line.
To define a normal line using a parametric equation, we use the point \( P_0 = (1, 1, -1) \) and the gradient vector \( (1, -3, -1) \):

• For x: \( x = 1 + t \)
• For y: \( y = 1 - 3t \)
• For z: \( z = -1 - t \)

The parameter \( t \) allows us to trace out the line across space as it passes through \( P_0 \) and extends in the direction determined by the normal vector.

Partial derivatives are a foundational concept in multivariable calculus. They measure how a function changes as only one of the input variables changes, holding all others constant. To compute the partial derivatives of a surface function, differentiate the function with respect to one variable at a time.
For the surface function \( f(x, y, z) = x^2 - xy - y^2 - z \):

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

These derivatives are essential for forming the gradient vector \( abla f \), which combines them into a single vector. This vector is used to find both the tangent plane and the normal line to the surface.

The surface equation is a mathematical way to describe a shape or a surface in a multi-dimensional space. For this exercise, the surface of interest is defined by the implicit equation \( x^2 - xy - y^2 - z = 0 \). This equation helps to understand how the surface behaves in the three-dimensional space formed by the x, y, and z axes.
The equation itself represents a set of all points \((x, y, z)\) that satisfy this condition, forming a surface. At any point \( P_0 \) on this surface, characteristics like the tangent plane and the normal line can be derived using the gradient vector.
The tangent plane at a point is formed using the gradient vector at that point, resulting in the equation: \( 1(x - 1) - 3(y - 1) - 1(z + 1) = 0 \), simplified to \( x - 3y - z + 3 = 0 \). This equation gives us a flat plane that just barely touches the surface at \( P_0 \), representing the surface's best linear approximation there.