Vector Differentiation
Lecture #3
Gradient
For a function f (x), what does df / dx imply? d f
d f d x
dx
f f
For a small change in x, how fast does f change ?
Consider a scalar function T (x, y, z)
x x
T T T
dT dx dy dz
x y z
dl xˆ d x yˆ d y zˆ d z
T T T ˆ
xˆ yˆ zˆ . x d x yˆ d y zˆ d z d T Geometric interpretation of Gradient:
x y z
For a given (dl) mag , dT is max when
= 0 T lies along which dT is max
T grad T xˆ yˆ zˆ
x y z
Consider a constant temperature surface:
Let
n̂ PQ = d l
T Q (x + x, y + y, z + z) xˆ d x yˆ d y zˆ d z
P (x, y, z)
dT ?
T . d l
Constant
temperature
Constant
surface
temperature
T l cos 2 = 0
surface
In cylindrical coordinates T (s, , z): T T T
dT ds d dz
s z
dl ds sˆ sd ˆ dz zˆ
dT T . d l
where T 1 T ˆ T
T s
ˆ zˆ
cyl s s z
In spherical coordinates (r, , ):
T T T
dT dr d d
r
dl dr rˆ rd ˆ r sin d ˆ
dT T . d l
T 1 T ˆ 1 T ˆ
T rˆ
sph r r r sin
xˆ yˆ zˆ
x y z
Vector ? T : Is this a product of del and T ?
: A vector operator
An ordinary vector can multiply in 3 different ways
Three different ways it can act on:
1. T on a scalar (via the gradient)
2. . v on a vector function (via the divergence)
3. v on a vector function (via the curl)
Curl operation + Fundamental
Theorems of Vector Calculus
vx v y vz
.v xˆ yˆ zˆ . v x xˆ v y yˆ v z zˆ
Recap:
x y z x y z
What would be div of a scalar? Meaningless! A scalar
Geometrical interpretation: Measure of how much v spreads out from the point P
P
P
P
P P
Positive divergence Negative divergence Zero divergence
A vector whose divergence is zero is also
A sphere embedded in a referred as solenoidal
3D-divergent vector field Example of a saw dust sprinkling at the
edge of a pond
.v in cylindrical coordinates:
1 1 v v z
.v svs
s s s z
In spherical coordinates
1 2 1 v
.v 2
r r
r vr 1
r sin
sin v
r sin
Curl of a vector: v
xˆ yˆ zˆ
zˆ v x xˆ v y yˆ v z zˆ
xˆ yˆ v a vector
z y z x y z
vx vy vz
Geometrical interpretation:
v is a measure of how much v curls/swirls around the point in question
For div v: z
z
P
y y
What will be its curl? x x
What will be the direction of the curl?
Example: drop a paddlewheel on the edge of a pond
y
x
Consider flow of water in a river and if we
wish to determine if it has a curl, then we put Calculate curls of
a paddle wheel into the water and see if it
turns i) va xyˆ
ii) vb yxˆ xyˆ
i)
xˆ yˆ zˆ ii) xˆ yˆ zˆ
v v
x y z x y z
0 x 0 y x 0
For a finite curl it will turn/spin ẑ 2 ẑ
otherwise it will not spin
In cylindrical coordinates (s,, z) :
1 v z v vs v z 1 vs
v sˆ sv
ˆ
zˆ
s z z s s s
In spherical coordinates (r, , ):
v 1 1 vr ˆ
v
1
sin v rˆ rv
r sin
r sin r
1 vr ˆ
r v
r r
Product rules:
f g f g g f
Involving gradient:
A. B A B B A A. B B . A
Involving divergence:
. f A f . A A . f
. A B B . A A. B
Involving curl:
f A f A A f
A B B . A A . B A . B B . A
T ?
T T T 2T 2T 2T
. T ˆ
x ˆ
y ˆ
z
.
ˆ
x ˆ
y ˆ
z
2
T
x y z x y z x y z
2 2 2
Laplacian of
In cylindrical coordinates: a scalar T
1 T 1 T T
2 2
T
2
s 2 2
s s s s 2
z
In spherical coordinates:
1 2 T 1 T 1 2T
T 2 r 2 sin 2 2 2
2
r r r sin r sin
T 0
. v 0
2
v . v v
Line, surface, and volume integrals:
y
b
Line integral:
a
v . dl
Path
dl b
a z
x
Physical example:
Work done by a force: W F . d l
If the Path forms a closed loop i.e. for b = a
v . d l
For a special class of vector functions, line integral is independent of the Path
True for conservative forces
Surface integral:
z da
v .d a
S
y
v .d a
For a closed surface:
(e.g. forming a balloon) x
If the vector function represents flow of a fluid, the surface integral will represent
total mass/time passing through the surface
Volume integral:
T d
v
a scalar
If T represents density of a substance, which may vary from point to point, then the
volume integral will represent total mass
Fundamental theorem for gradients:
Consider a scalar function T (x, y, z)
y
Starting at a as one moves a short distance dl1 change in
b T will be
d T T . d l1
dl1
a z Total change in T as one moves from a to b along a
specified path:
x b b
Path
a
T . d l dT T (b) T (a )
a
v .dl 0
This is the algebraic form of the fundamental theorem for gradients
Line integral of the derivative of a function is given by the value of the function at the
boundaries
Example: measuring height of Eiffel Tower
b b
Path
a
T . d l dT T (b) T (a )
a
1. Measure each step and add LHS
2. Difference between two
readings of an altimeter RHS
• Height: 1063 ft (324 m)
• Steps: 1710
b
Corollary 1: T . d l is independent of the path from a to b
a
Line integral of gradient is path independent
T b T b 0
Corollary 2:
T . d l 0
Fundamental theorem for divergences :
. v d v . d a
V
S
Also known as
Gauss’s theorem or Green’s theorem or divergence theorem
Integral of the divergence of a vector function over a region
equals the value of the function at the boundary i. e. surface that bounds the volume
Geometrical interpretation:
If v: flow of an incompressible liquid then the R.H.S would represent total
amount of liquid passing out/time through the surface
Divergence means spreading out e.g. Tap pouring out water
For a bunch of taps assuming each pours out equal amount of
incompressible fluid, one can determine total amount by using the
divergence theorem
(taps within the volume) = (flow of fluid out through the surface)
Fundamental theorem for curls:
v . da v . d l
S P
Integral of a derivative of a function over a region equals value of the function at its boundary
Curl Perimeter
Geometrical interpretation:
Twist of v
v . da
S
v .dl
P
Corollary 1: v . d a Depends only on the boundary line and not on the surface
v . d a 0
Corollary 2: For a close surface e.g. mouth of a balloon
