1. Tangent Planes and Normal Lines

2. 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.

3. Example-1

4. 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

5. What is vector-valued function?

8. 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

9. 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

10. Or
0 =

12. 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.

13. Tangent Plane and Normal Line

14. Normal Line

15. 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

17. 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

18. Symmetric Equation of a normal Line

19. Example-2:

21. For Animation

22. 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.”

23. 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.

24. 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.

25. Do you remember how to find angle between two planes?

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

27. Class-work-1

29. Class-work-2

30. Class-work-3

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