CSCI 2170 Lecture Notes - Lecture 15: Iterative Method, Surface Integral, Divergence Theorem
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Math 241 Chapter 15 Dr. Justin O. Wyss-Gallifent
§15.1 Vector Fields
1. Define a vector field: Assigns a vector to each point in the plane or in 3-space. Can be visualized
as loads of arrows. Can represent a force field or fluid flow - both are useful.
2. Two important definitions. Often before I do these I define ∇=∂
∂x ˆı+∂
∂y ˆ+∂
∂z ˆ
kso that gradient,
divergence and curl all make sense with how ∇is used.
(a) The divergence ∇ · ¯
F=Mx+Ny+Pzgives the net fluid flow in/out of a point (very small
ball).
(b) The curl ∇ × ¯
Fgives the axis of rotation of the fluid at a point.
3. For a function fwe saw the gradient ∇fis a VF. In fact it’s a special kind of VF. Any VF which
is the gradient of a function fis conservative and the fis a potential function.
There are two facts to note:
(a) If ¯
Fis conservative then ∇× ¯
F=¯
0 and consequently if ∇× ¯
F6=¯
0 then ¯
Fis not conservative.
Moreover if ∇ × ¯
F=¯
0 and ¯
Fis defined for all (x, y, z) then ¯
Fis conservative.
(b) If we have ¯
Fwe can tell if it’s conservative by the above method and we can find the potential
function too using the iterative method. Make sure to do 2-variable and 3-variable cases.
§15.2 Line Integrals (of Functions and of VFs)
1. If Cis a curve and fgives the density at any point then we can define the line integral of fover/on
C, denoted RCf ds, as the total mass of C. We evaluate it by parametrizing Cas ¯r(t) on [a, b] and
then RCf ds =Rb
af(x(t), y(t), z(t))||¯r′(t)|| dt. The result is independent of the parametrization
and the orientation.
Sample units: Cin cm, fin g/cm and the result in g.
2. If Cis the path of an object through a force field ¯
Fthen we can define the line integral of ¯
F
over/on C, denoted RC¯
F·d¯r, as the total work done by ¯
Fas it traverses C. The most basic way to
evaluate it is by parametrizing Cas ¯r(t) on [a, b] and then RC¯
F·d¯r=Rb
a¯
F(x(t), y(t), z(t))·¯r′(t)dt.
Some notes about line integrals of vector fields:
(a) The orientation (direction) of Cmatters. If −Cis the same curve in the opposite direction
then R−C¯
F·d¯r=−RC¯
F·d¯r. This makes sense for work done.
(b) The parametrization in that direction doesn’t matter.
(c) There is alternate notation for this integral. We can write RCM dx +N dy +P dz which
means the same as RC(Mˆı+Nˆ+Pˆ
k)·d¯r. Watch out for things like RCM dx which looks
deceivingly like a regular integral.
Sample units: Cin cm, ¯
Fin g ·cm/s (dynes) and the result in g ·cm2/s2(ergs).