Tangent Planes

Let

• is a 3-variable function
• is the level surface of at
• be a continuous curve on
• be a point on

All points on satisfy the equation . The below equation can be deduced by differentiating the equation.

The gradient vector of is perpendicular to the tangent vector (of ).

Tangent plane to can be defined by applying the above equation at point . If , then the tangent plane to at is the one that passes through and has a normal vector .

The equation of the tangent plane:

Normal line

Normal line to the surface at point , is the line passing through and perpendicular to the tangent plane. The equation of the normal line is: