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)
Fundamental theorem for gradients
In vector calculus, there are three derivatives (gradient, divergence, and curl),
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)
Now, calculate the same integral along path (iii):
Are they consistent with the fundamental theorem?
Yes! T(b) – T(a) = 2 – 0 = 2
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).
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
Gauss's theorem
Green's theorem
Divergence theorem
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).
Example 1.11
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
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: ( , , ) u v w
A.2 Notation
1.4.1 Spherical Polar Coordinates
sin cos sin sin cos x r y r z r
( , , ) r
polar angle
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 Coordinates ( , , ) r
1.4.2 Cylindrical Coordinates
cos sin x s y s z z
( , , ) s z
azimuthal angle