Calculus III:

Tangent vectors
Normal vectors

We are going to determine the tangent and normal vectors to level surfaces and parametric surfaces.

A level surface is defined by the equation: f(x,y,z) = k, while parametric surface is defined by the position vector (u,v).

1. Tangent and normal vectors
to level surfaces

Consider a level surface defined by f(x,y,z) = k. The surface is said smooth if its gradient ∇f is continuous and non-zero at any point on the surface.

∇f = f_x + f_y + f_z

Let's suppose that (t) = 〈 x(t), y(t), z(t)〉 lies on the smooth surface f(x,y,z) = k.

Applying the derivative with respect to t to both sides of the equation of the level surface yields:

df/dt = dk/dt

k is a constant, the chain rule implies that

df/dt =
(∂f/∂x)(∂x/∂t) + (∂f/∂y)(∂y/∂t) + (∂f/∂xz)(∂z/∂t) =
= ∇f . = dk/dt = 0

Where = 〈 x', y', z' 〉 is the tangent vector to the surface at the point (x,y,z).

∇f . = 0 implies that the two vectors ∇f and are perpendicular. So ∇f is orthogonal to each tangent vector at a given point on the surface.

Therefore

At a given point (x,y,z), = ∇f(x,y,z) is normal to the surface f(x,y,z) = k at the point (x,y,z).

Let's remember that:

The gradient ∇f is normal to the surface f(x,y, z) = k at each point on the surface.

Exercice

Show, otherwise, that ∇f is orthogonal to the level curve f(x,y,z) = k.

Let's consider a two-variables function f(x,y) and then we will do an extension for the three-variable function f(x,y,z).

The differential of f(x,y) = k gives

df = (∂f/∂x)dx + (∂f/∂y)dy = dk = 0

That can be written as a dot product:

(∂f/∂x)dx + (∂f/∂y)dy = 〈f_x,f_y〉 . 〈dx, dy〉 =
∇f . = 0

That is ∇f and are orthogonal.



We know that the vector is tangent to the curve. So ∇f is normal to the curve.

The work for the level curve f(x,y,z) = k is identical. Then

The gradient ∇f is normal to the surface f(x,y,z) = k at each point on the surface.

2. Tangent and normal vectors
to parametric surfaces

If we parameterize the fonction f(x,y,z) by the vector
(u,v), then
= ∂/∂u , and
= ∂/∂v
are tangent to the surface.

Therefore their cross product × is normal to the surface.

Let's remember that:

= × is normal to the surface.