1. Chapter 9
Vector Calculus

2. Outline
9.1 Vector Functions 9.2 Motion on a Curve 9.3 Curvature and Components of Acceleration 9.4 Partial Derivatives 9.5 Directional Derivative 9.6 Tangent Planes and Normal Lines 9.7 Curl and Divergence 9.8 Line Integrals 9.9 Independence of the Path

3. Outline (cont’d.)
9.10 Double Integrals 9.11 Double Integrals in Polar Coordinates 9.12 Green’s Theorem 9.13 Surface Integrals 9.14 Stokes’ Theorem 9.15 Triple Integrals 9.16 Divergence Theorem 9.17 Change of Variables in Multiple Integrals

4. Vector Functions
A vector function r has components that are functions of a parameter t
For a given value of t, r is the position vector of a point P on a curve C

5. Vector Functions (cont’d.)
If limits of the component functions exist,
For all t for which the limit exists,
If the components are differentiable,
If the components are integrable,

6. Vector Functions (cont’d.)
From the chain rule, where is a differentiable scalar function,

7. Motion on a Curve For a body moving along a curve,
For circular motion in a plane
For projectiles,
position
velocity
acceleration
position
centripetal acceleration

8. Curvature and Components of Acceleration
Curvature of C at a point is
where
The radius of curvature  is 1/
 at a point P on a curve C is the radius of a circle that best fits the curve
The circle is the circle of curvature
Its center is the center of curvature

9. Partial Derivatives If the partial derivative with respect to x is
and with respect to y is

10. Directional Derivative
Gradient of a function is
Consider

11. Tangent Plane and Normal Lines
The normal line to a surface at P

12. Curl and Divergence
Vector functions of two and three variables are also called vector fields

13. Line Integrals

14. Line Integrals (cont’d.)

15. Independence of Path
A piecewise smooth curve C between an initial point A and a terminal point B is a path
Line integrals involving a certain kind of field F depend not on the path both only on the endpoints

16. Independence of Path (cont’d.)

17. Double Integrals

18. Double Integrals (cont’d.)
Type I Region
Type II Region

19. Double Integrals in Polar Coordinates

20. Green’s Theorem
The double integral over the region R bounded by a piecewise smooth curve C gives the area of the bounded region

21. Surface Integrals
Surfaces integrals may be used to evaluate mass of a surface, flux through a surface, charge on a surface, etc.

22. Stokes’ Theorem
Stokes’ Theorem is the three-dimensional form of Green’s Theorem

23. Stokes’ Theorem (cont’d.)

24. Triple Integrals Triple integral applications Volume of solids
Mass of solids
First and second moments of solids
Coordinates of center of mass
Centroid of solids

25. Divergence Theorem
The Divergence Theorem is useful in the derivation of some of the famous equations in electricity and magnetism and hydrodynamics

26. Change of Variables in Multiple Integrals
At times, it is either a matter of convenience or necessity to make a change of variable in a definite integral in order to evaluate it (to simplify the integrand or region of integration)
The transformation of triple integrals follows suit