Final answer:
To find the unit vector normal to the surface 5z² - x⁴, compute the gradient of the scalar function and normalize it. The gradient vector is (-4x³, 0, 10z), which after normalization gives the unit vector normal to the surface.
Step-by-step explanation:
To find a unit vector n that is normal to the surface described by the equation 5z² - x⁴ = 0, we need to take the gradient of the scalar function. The gradient provides a vector that points in the direction of the greatest rate of increase of the function, which would be perpendicular to the level surfaces.
The partial derivatives of 5z² - x⁴ are:
- with respect to x: -4x³
- with respect to y: 0 because y does not appear in the equation
- with respect to z: 10z.
The gradient vector is thus ∇(5z² - x⁴) = (-4x³, 0, 10z). However, this is not necessarily a unit vector. To find the unit normal vector, we divide this gradient vector by its magnitude to normalize it.
The magnitude of the gradient is given by √((-4x³)² + 0² + (10z)²). This normalized vector is our desired unit vector n.