Tangent Planes and Normal Lines

TANGENT PLANES to Level Surfaces Suppose S is a surface with equation z= f(x,y) Can it be written as f(x,y)-z = 0? How about F(x, y, z) = 0? Hence, it is a level surface of a function F of three variables.

Example-1

Now consider the level surface, S: F(x, y, z) = 0 Let P(x0, y0, z0) be a point on S. and let C be a curve on S through P that is defined by the vector-valued function r(t) = x(t)i + y(t)j + z(t)k Then for all t, F(x(t), y(t), z(t) ) = 0

What is vector-valued function?

Difference between real-valued function f & g and vector-valued function, r? * All are functions of the real variable t ** we use the vector-valued function to represent the motion along a curve Or to trace the graph of a curve

If x, y, and z are differentiable functions of t and F is also differentiable, then we can use the Chain Rule to differentiate both sides of F(x(t), y(t), z(t) ) = 0

Or 0 =

The equation Implies that: “The gradient vector at P, is orthogonal (perpendicular) to the tangent vector r’(t0) to any curve C on S that passes through P.

Tangent Plane and Normal Line

Normal Line

What will be the equation of the tangent plane to S at P(x0, y0, z0) ? Let (x, y, z) be arbitrary point on the tangent plane. Does this vector lies on the tangent plane? Any comment on

Normal Line The normal line to S at P is the line: Passing through P Perpendicular to the tangent plane Thus, the direction of the normal line is given by the gradient vector

Symmetric Equation of a normal Line

Example-2:

For Animation http://www.math.umn.edu/~rogness/multivar/conenormal.html http://www.math.umn.edu/~rogness/math2374/paraboloid_normals.html http://www.math.umn.edu/~rogness/multivar/tanplane_withvectors.shtml

One day I saw Chelsea during the morning exercise, spent the entire period standing leaning at about a 30 degree angle from standing up straight. I asked her “Why are you not standing up straight? “ She replied “Sorry, I am not feeling normal.”

The angle Inclination of a plane Another use of the gradient F(x, y, z) is to determine the angle of inclination of the tangent plane to a surface.

The angle Inclination of a plane The angle of inclination of a plane is defined to be the angle between the given plane and the xy-plane.

Do you remember how to find angle between two planes?

Let n be the normal vector to the given plane. What is the normal vector to the xy-plane?

Class-work-1

Class-work-2

Class-work-3

Home-work-4 Find the equations of the tangent plane and normal line at the point (–2, 1, –3) to the ellipsoid