# 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).