Chapter 1. Vector Analysis 1.3 Integral Calculus 1.3.1 Line, Surface, and Volume Integrals (a) Line Integrals. A line integral is an expression of the form

If the path P in question forms a closed loop (that is, if b = a),

Example 1.6

(path 1)

(path 2)

For the loop that goes out (1) and back (2)  Line, Surface, and Volume Integrals

(b) Surface Integrals. A surface integral is an expression of the form perpendicular to the surface

If the surface is closed

Example 1.7 Line, Surface, and Volume Integrals

(c) Volume Integrals. A volume integral is an expression of the form

A vector functions

Example 1.8 1.3.2 The Fundamental Theorem of Calculus

OR

Fundamental theorem  the integral of a derivative over an interval is given by the boundaries (the value of the function at the end points)

In vector calculus, there are three derivatives (gradient, divergence, and curl),

Fundamental theorem for gradients

Fundamental theorem for divergences

 Gauss's theorem, Green's theorem, or, simply, the divergence theorem

Fundamental theorem for curls

 Stokes’ theorem 1.3.3 The Fundamental Theorem for Gradients

The integral (here a line integral) of a derivative (here the gradient) is given by the value of the function at the boundaries (a and b).

Example 1.9 Check the fundamental theorem for gradients.

 Let's go out along the x axis, step (i), and then up, step (ii) Are they consistent with the fundamental theorem?  Yes! T(b) – T(a) = 2 – 0 = 2

 Now, calculate the same integral along path (iii): 1.3.4 The Fundamental Theorem for Divergences

The integral (here a volume integral) of a derivative (here the divergence) is given by the value of the function at the boundaries (surface).

 Gauss's theorem  Green's theorem  Divergence theorem

If v represents the flow of an incompressible fluid, then the flux of v is the total amount of fluid passing out through the surface, per unit time.

Example 1.10 Check the divergence theorem using the function

 In this case,

 To evaluate the surface integral,

~ 1.3.5 The Fundamental Theorem for Curls

The integral (here a surface integral) of a derivative (here the curl) is given by the value of the function at the boundaries (perimeter of the surface).

 Stokes’ theorem

For dl, which way are we supposed to go around (clockwise or counterclockwise)? For da, which way does it point? which way is "out?"  Consistency in Stokes' theorem  Let’s Keep the right-hand rule

Example 1.11 1.3.6 Integration by Parts

Integrating both sides

 We can transfer the derivative from g to f,  at the cost of a minus sign and a boundary term.

Example 1.12

Note:

by the product rules of vector calculus

From the divergence theorem  Same as the integration by parts 1.4 Curvilinear Coordinates A.1 (orthogonal) Curvilinear Coordinates: (,,uvw )

A.2 Notation 1.4.1 Spherical Polar Coordinates (,r  , )

polar angle xr sin cos yr sin sin zr cos  azimuthal angle

Beware that The unit vectors, , at a particular point P, change direction as P moves around.  Do not naively combine the spherical components of vectors associated with different points

The unit vectors themselves are functions of position  Do not take outside the integral Spherical Polar Coordinates (,r  , )

Infinitesimal displacement  Infinitesimal volume 

Infinitesimal surfaces  depend on the orientation of the surface r is constant   is constant 

Example 1.13 Find the volume of a sphere of radius R. Vector derivatives in (, r  ,  ) Coordinates 1.4.2 Cylindrical Coordinates (,s  ,z )

xs cos ys sin zz

azimuthal angle Appendix A: Vector Calculus in Curvilinear Coordinates

fg h x, y, z 1 1 1 s, , z 1r 1 r, ,  1rr sin