1. Tangent Vectors and Normal Vectors
Unit Tangent Vector: Let be a smooth curve represented by on an open interval The unit tangent vector at is defined to be

2. Example: (Larson, Example-1,Page-807) Find the unit tangent vector to the curve given by , when
Solution: Given,
Thus, the unit tangent vector is
When , the unit tangent vector is
Ans.

3. Principal Unit Normal Vectors: Let be a smooth curve represented by on an open interval . If then the principal unit normal vector at is defined to be

4. Example: (Larson, Example-3,Page-809) Find principal unit normal vector for the curve represented by ,when
Solution: (Do)

5. Example: (Larson, Example-4,Page-810) Find the principal unit normal vector for he helix given by , when
Solution:

6. Thus, the principal unit normal vector is
When , the principal unit normal vector is
Ans.

7. Tangential and Normal Components of Acceleration: If is the position vector for a smooth curve [for which
exists], then the tangential and normal components of acceleration are as follows.
Example: Find the tangential and normal components of acceleration for the position function given by

8. Solution: Given, The tangential component of acceleration is
And the normal component of acceleration is

9. Practice Problems Problems: 14,17,18;29,30; Exercises for Section 12.4
Book: Calculus with Analytical Geometry (5th ed.)
(Larson, Edward and Hostetler)
Exercises for Section 12.4
Page no: 814
Problems: 14,17,18;29,30;

10. Arc Length and Curvature
Arc Length: The arc length of a curve given by is

11. Example: (Larson, Example-3,Page-817) Find the arc length of one turn of the helix given by
Solution: Here,
Thus, the arc length of one turn of the helix is
Ans.

12. Curvature: The curvature measures how fast a curve is changing direction at a given point.
Let be a smooth curve given by ,where is the arc length parameter. The curvature at is given by
[Formal definition]
Let be a smooth curve given by ,then the curvature at is given by
[General definition]

13. where kappa=curvature
Mathematically
Suppose that a particle moves along the curve with unit speed. Taking the time s as the parameter for C, this provides a natural parametrization for the curve. The unit tangent vector T (which is also the velocity vector, since the particle is moving with unit speed) also depends on time. The curvature is then the magnitude of the rate of change of T. Symbolically,
where kappa=curvature

14. Problems
Problems-14(826): Find the curvature of the curve where s is the arc length parameter . Similar problems: Ex-5(820),13(826)

15. Solution: Given vector valued function
Example: (Larson, Example-7,Page-821) Find the curvature of the curve given by
Solution: Given vector valued function
We have, Tangent vector

17. We know, Curvature
Ans.

18. Practice Problems Problems: 21, 23,29,32; Problem 1:
Find the arc length of the curve
on the interval
Book: Calculus with Analytical Geometry (5th ed.)
(Larson, Edward and Hostetler)
Exercises for Section 12.5
Page no: 825
Problems: 21, 23,29,32;

19. Directional Derivatives and Gradients
To determine the slope at a point on a surface, we define new type of derivative called a Directional Derivative. And to determine in which direction at that point on that surface the slope is maximum, we introduce Gradient.

20. Definition of Directional derivative: Let be a function of two variables and in the direction of the unit vector
Then the directional derivative of in the direction of denoted by and is defined by
[difficult to use] Or
[simpler than previous one]
[for 3 dimensions]

21. Example: (Larson, Example-1, Page-879) Find the directional derivative
of at in the direction of
Solution: Given function
Given direction unit vector,

22. We have,
Therefore the directional derivative at is
Ans

23. Example: (Larson, Example-2, Page-879) Find the directional derivative
of at in the direction of
Solution: Given function
Given direction vector,
direction unit vector

24. We have,

25. Definition of Gradient: Let be a function of two variables
and such that and exist. Then the gradient of is denoted by and is defined by the vector
[ for two variables]
[ for three variables]

26. Properties of the Gradient:

27. Example: (Larson, Example-8, Page-885) Find the gradient of
at the point
Solution: Given function
We have,

28. Problems
Ex-5.The temperature in degree celcius on the surface of metal plate is Where x and y are measured in inches. In what direction from (2,-3) does the temperature increase most rapidly? What is the rate of increase?

29. Solution: y X

30. Practice Problems Problems: 1,5,7,12; 17,20; 25,27,30;
Book: Calculus with Analytical Geometry (5th ed.)
(Larson, Edward and Hostetler)
Exercises for Section 13.6
Page no: 886
Problems: 1,5,7,12; 17,20; 25,27,30;

31. Tangent Planes and Normal Lines
One nice use of tangent planes is they give us a way to approximate a surface near a point. Normal lines are important in analyzing surfaces and solids.
Surface S:
Normal Line
Tangent
Plane

32. Definition of Tangent Plane and Normal Line:
Let be differentiable at the point on the surface given by such that
The plane through that is normal to is called the tangent plane to at In particular the equation of the tangent plane is

33. 2. The line through having the direction of is called the normal line to at In particular the equation of the normal line is

34. Example: (Larson, Example-2, Page-891) Find an equation of the tangent plane to the hyperboloid given by at the point
Solution: Given equation is

35. at the point , the partial derivatives are
We have, equation of tangent line at is

36. Example: (Larson, Example-3, Page-892) Find an equation of the tangent plane to the paraboloid given by at the point
Solution: Do

37. Solution: Given equation of surface is
Example: (Larson, Example-4, Page-893) Find the equation of the normal line to the surface given by at the point
Solution: Given equation of surface is
Let

38. at the point , we get
we have, the parametric equation of normal line at is

39. Practice Problems Problems: 11,17,21;26,29; Exercises for Section 13.7
Book: Calculus with Analytical Geometry (5th ed.)
(Larson, Edward and Hostetler)
Exercises for Section 13.7
Page no: 895
Problems: 11,17,21;26,29;

40. Vector Fields
Functions that assign a vector to a point in the plane or to a point in space. Such functions are called vector fields, and they are useful in representing various types of force fields and velocity fields.
Definition:
Let and be functions of two variables and , defined on a plane region The function defined by
is called a vector field over
and is called a vector field over
Vector Field

41. Conservative Vector Field:
A vector field is called conservative if there exists a differentiable function such that The function is called the potential function for .
Example: (Larson, Example-4, Page-996) Show that the vector field given by is conservative if there exists
Solution: Given
Because
it follows that is conservative. ( Showed)

42. Test for Conservative Vector Field in Plane:
Let and have continuous first partial derivatives on an open disc The vector field given by is conservative if and only if
Example: (Larson, Example-5, Page-997) Show that the vector field given by is not conservative.
Solution: Here, and
Since , it shows that is not conservative.
(Showed).

43. Example: (Larson, Example-6, Page-998) Find a potential function for
Solution: Given function
Since , so that is conservative
comparing (i) and (ii), we get

44. Integrating (iii) with respect to x, we obtain
[ Where g(y) is constant with respect to x. ]
Integrating (iv) with respect to y, we obtain
[ Where h(x) is constant with respect to y. ]
Therefore the required potential function is
Ans.

45. Example: (Larson, Example-8, Page-1000) Find a potential function for
Solution: Do.

46. Practice Problems Problems: 21,27;37,39; Exercises for Section 15.1
Book: Calculus with Analytical Geometry (5th ed.)
(Larson, Edward and Hostetler)
Exercises for Section 15.1
Page no: 1002
Problems: 21,27;37,39;

47. Curl of a Vector Field
Curl of a vector field captures the idea of how a fluid may rotate. Imagine that the below vector field  represents fluid flow. The vector field indicates that the fluid is circulating around a central axis.

48. Definition of Curl of a vector field:
The Curl of a vector field is
Remark: If , then we say that is irrotational and conservative.

49. Example: (Larson, Example-6, Page-998) Find the curl of a Vector Field
Solution: The curl of a Vector Field is given by

50. Practice Problems Problems: 29,31; Exercises for Section 15.1
Book: Calculus with Analytical Geometry (5th ed.)
(Larson, Edward and Hostetler)
Exercises for Section 15.1
Page no: 1002
Problems: 29,31;

51. Divergence of a Vector Field
Imagine that the vector field F  pictured below gives the velocity of some fluid flow. It appears that the fluid is exploding outward from the origin (fig-1).This expansion of fluid flowing with velocity field F is captured by the divergence of  F, which we denote divF.
The divergence of that vector field is positive since the flow is expanding.
Since this compression of fluid is the opposite of expansion in (fig-2), the divergence of this vector field is negative.
Fig-1
Fig-2

52. Definition of Divergence of a Vector Field
The divergence of is denoted by and defined by
If , then is said to be divergence free.

53. Example: Find the divergence at for the vector
field
Solution: The divergence of
at the point , the divergence is

54. Practice Problems Problems: 49,52,53; Exercises for Section 15.1
Book: Calculus with Analytical Geometry (5th ed.)
(Larson, Edward and Hostetler)
Exercises for Section 15.1
Page no: 1002
Problems: 49,52,53;

55. End (of Mid Term Syllabus)

56. Discussed Topics Tangent Vectors and Normal Vectors
Arc Length and Curvature
Directional Derivatives and Gradients
Tangent Planes and Normal Lines
Vector Fields