The equation of the paraboloid is given by z = x² + y². This represents a surface that opens upwards, with its vertex at the origin (0, 0, 0).
To sketch the graph, we can visualize it in three dimensions. The contour lines on the x-y plane are circles centered at the origin, with radius increasing as z increases. Each slice of the paraboloid at a constant z will be a circle, demonstrating the symmetrical nature of this surface.
Now, to determine the outward normal vector n, we first need to find the gradient of the function f(x, y, z) = z – x² – y². The gradient will give us the normal vector to the surface defined by this equation.
The gradient is calculated as follows:
∇f = (∂f/∂x, ∂f/∂y, ∂f/∂z)
= (-2x, -2y, 1)Thus, the normal vector at any point (x, y) on the surface is given by:
n = (-2x, -2y, 1)To determine the orientation of this normal vector, we can evaluate its direction. Since the paraboloid opens upwards, we want the normal vector to point outwards from the surface. This means that the z-component of our normal vector, which is 1, is indeed in the positive k direction.
In summary, the outward normal vector n at any point (x, y) on the surface z = x² + y² can be expressed as:
n = (-2x, -2y, 1)and it points in the +k direction (upwards), as expected for a surface that opens upward.