Vector Calculus & General Coordinate Systems

Vector Calculus & General Coordinate Systems

Derivatives of a General Unitary Basis

We have examined the computation of derivatives in an
orthonormal curvilinear system. With the exception of the
Cartesian system, these are the easiest to compute. Now we
will develop more general procedures and formulae for an
arbitrary curvilinear system with a unitary basis. Although the
development is straightforward, we will introduce symbols and
nomenclature that is probably unfamiliar. You must pay close
attention to the definitions.

Computational mechanics (fluid, structures, etc.) when the
physical domain is mapped onto a curvilinear grid is an
example where the basis vectors change with respect to
location in the grid, thus partial derivatives such as ∂ei / ∂q j
must be evaluated.

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Christoffel symbols and contravariant derivatives
Recall, for an arbitrary vector a,

a = (a ⋅ ei )ei
If a = ∂ei / ∂q j, we can write

∂ei = ⎛ ∂ei ⋅ek ⎞ ek . (11)
∂q j ⎜⎝ ∂q j ⎟
⎠

For a more compact notation, let’s define the Christoffel
symbol of the second kind,

⎧ k ⎫ = ∂ei ⋅ ek .
⎨⎩i ⎬ ∂q j
j ⎭ (12)

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Now, the contravariant derivative of the covariant basis is,

∂ei = ⎧ k ⎫ ek . (13)
∂q j ⎨⎩i ⎬
j ⎭

Note that the christoffel symbol is the contravariant component
of the ek basis. In eq. (13) note the summation over the dummy
k. Reference to the dual basis (ek) is eliminated by introducing

the components of the fundamental metric,

ek = g krer ,

∂ei ⋅ (ek = g krer ),
∂q j

⎧ k ⎫ = g kr ∂ei .
⎨⎩i ⎬ ∂q j
j ⎭ (14)

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Vector Calculus & General Coordinate Systems

A few more manipulations will enable us to write the
Christoffel symbol completely in terms of components of the
fundamental metric. First, rewrite eq. (14),

⎧ k ⎫ = 1 g kr ⎛ ∂ei ⋅ er + ∂ei ⋅ e ⎞
⎨⎩i ⎬ 2 ⎝⎜ ∂q j ∂q j ⎟.
j ⎭ r ⎠ (15)

Now use the fact that r(qi) is a continuous function then,

∂ei = ∂ ⎛ ∂r ⎞ = ∂ ⎛ ∂r ⎞ = ∂e j . (16)
∂q j ∂q j ⎝⎜ ∂qi ⎠⎟ ∂qi ⎜⎝ ∂q j ⎟⎠ ∂qi

This result might be unexpected. The equality of mixed
second-partial derivatives, however, should not surprise us if
we recall from the differential calculus,

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∂2 f = ∂2 f
∂x∂y ∂y∂x

iff f (x,y) is continuous. Substituting eq. (16) into eq. (15), we
have,

⎧k ⎫ = 1 g kr ⎛ ∂e j ⋅ er + ∂ei ⋅ er ⎞ .
⎩⎨i ⎬ 2 ⎜ ∂qi ∂q j ⎟
j ⎭ ⎝ ⎠ (17)

For the final steps of this development, we now write the dot
products in terms of the fundamental metric components,

∂ (e j ⋅ ei ) = ∂e j ⋅ er + e j ⋅ ∂er ,
∂qi ∂qi ∂qi

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Vector Calculus & General Coordinate Systems

or

∂e j ⋅ er = ∂ (e j ⋅er ) − e j ⋅ ∂er .
∂qi ∂qi ∂qi

Then,

⎧ k ⎫ = 1 g kr ⎡∂ (e j ⋅er ) − e j ⋅ ∂er + ∂ (ei ⋅ er ) − ei ⋅ ∂er ⎤
⎩⎨i ⎬ 2 ⎣⎢ ∂qi ∂qi ∂q j ∂q j ⎦⎥ ,
j ⎭

⎧ k ⎫ = 1 g kr ⎡∂ (e j ⋅ er ) + ∂ (ei ⋅ er ) − ∂ (ei ⋅ e ) ⎤ ,
⎩⎨i ⎬ 2 ⎣⎢ ∂qi ∂q ∂q ⎥
j ⎭ j r j ⎦

or finally,

⎧ k ⎫ = 1 g kr ⎡ ∂g jr + ∂gir − ∂gij ⎤
⎨⎩i ⎬ 2 ∂q j ∂qr ⎥.
j ⎭ ⎢ ∂qi ⎦ (18)
⎣
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Vector Calculus & General Coordinate Systems

Thus, eq. (18) gives the scalar components of the covariant
derivative in eq. (13).

Now we develop a similar relation for the contravariant
derivative of the contravariant basis. Start with,

∂ (ei ⋅ek ) = ∂ δ k = 0,
∂q j ∂q j i

∂ei ⋅ ek + ei ⋅ ∂ek = 0,
∂q j ∂q j

ei ⋅ ∂ek = − ∂ei ⋅ ek = − ⎧k j ⎫ (19)
∂q j ∂q j ⎩⎨i ⎬.
⎭

Again employ the relation a = (a ⋅ei )ei.

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Vector Calculus & General Coordinate Systems

If a = ∂ek / ∂q j , we can write

∂ek = ⎛ ∂ek ⋅ ei ⎞ ei ,
∂q j ⎜ ∂q j ⎟
⎝ ⎠

or, using eq. (19),

∂ek = − ⎧k j ⎫ ei . (20)
∂q j ⎩⎨i ⎬
⎭

Again, eq. (20) is the contravariant derivative of the
contravariant basis.

Finally, without going through the details, one can show,

⎧k ⎫ = ∂ ln g = 1 ∂g . (21)
⎩⎨i k ⎬ ∂qi 2g ∂qi
⎭

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Vector Calculus & General Coordinate Systems

Christoffel symbol of the first kind
The Christoffel symbol of the first kind, is defined in terms of
derivatives of the transformation relating the curvilinear
system to the original Cartesian system,

[ij, m] = ∂r ⋅ ∂r = e m ⋅ ∂ei . (22)
∂qm ∂qi∂q j ∂q j

Now with a = (a ⋅ em )em ,

∂ei = ⎛ ∂ei ⋅ em ⎞ em , (23)
∂q j ⎜⎝ ∂q j ⎟
⎠ (24)

or 183

∂ei = [ij, m]em.
∂q j

Vector Calculus & General Coordinate Systems

The difference between eqs. (23) and (24) and eqs. (12) and
(13) is that eqs. (12) and (13 ) give the contravariant derivative
of the covariant basis in terms of contravariant scalar
components and the covariant basis. Equations (23) and (24)
give the contravariant derivative in terms of covariant scalar
components and the contravariant (dual) basis.

To summarize and show the relation of the two Christoffel
symbols,

∂ei = ⎧k j ⎫ ek Expansion in terms of ek
∂q j ⎨⎩i ⎬ Expansion in terms of ek
⎭

∂ei = [ij, k]ek
∂q j

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Vector Calculus & General Coordinate Systems

The symbols can be directly related with ek = gkmem,

∂ei = [ij, m]em = ⎧k j ⎫ gkmem . (25)
∂q j ⎨⎩i ⎬
⎭

Now we can write [ij,m] in terms of the covariant components
of the fundamental metric,

[ij, m] = gkm ⎧ k ⎫ = 1 ⎡ ∂gim + ∂g jm − ∂gij ⎤ . (26)
⎩⎨i j ⎭⎬ 2 ⎢⎣ ∂q j ∂qi ∂qm ⎦⎥

Example: Orthogonal curvilinear coordinates

For orthogonal curvilinear coordinates, recall,

gij = 0, i ≠ j hi gii = 1 (no summation)
gii

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Vector Calculus & General Coordinate Systems

⎧ 1⎫ = 1 ⎡ ⎛ ∂g11 + ∂g11 − ∂g11 ⎞ + g12 ⎛ ∂g12 + ∂g12 − ∂g11 ⎞
⎩⎨1 1⎭⎬ 2 ⎢ g11 ⎜ ∂q1 ∂q1 ∂q1 ⎟ ⎜⎝⎜ ∂q1 ∂q2 ∂q2 ⎟⎟⎠
⎢⎣ ⎝ ⎠

+ g13 ⎛ ∂g13 + ∂g13 − ∂g11 ⎞⎤
⎜⎝⎜ ∂q1 ∂q2 ∂q3 ⎟⎠⎟⎥⎥⎦

= 1 1 ⎛ ∂g11 − ∂g11 ⎞ = 1 1 ∂(h12 )
2 g11 ⎜2 ∂q1 ∂q1 ⎟ 2 h12 ∂q1
⎝ ⎠

= 1 ∂h1 .
h1 ∂q1

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Vector Calculus & General Coordinate Systems

These results of course agree with eqs. (25) and (26).
Before concluding this discussion, it is worth mentioning that
the Christoffel symbol of second kind is most often used in the
literature. It is often denoted with the symbol, Γijk .

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Vector Calculus & General Coordinate Systems

Directional Derivative, Gradient, and Divergence

Directional derivative and gradient

Imagine a day in the Boulder area with a temperature
distribution (°F) shown in the contour schematic. With With
Boulder at the origin of the 2-D coordinate system, the
temperature distribution is an example of a scalar field, i.e., a
scalar function of a vector (in this case, the position).

Lyons 87 Lmt
Ned Nwt

87 B89ldr Laf 88
90 90 Lou

Sup Bfd

89 Wmr
US-36

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Vector Calculus & General Coordinate Systems

Directional Derivative, Gradient, and Divergence

In general,

φ = φ(r) = φ(q1, q2, q3) (φ = temperature) .

Suppose we want to determine the rate of change of φ roughly

along US-36 towards Denver, indicated by the eˆ direction. To
do this, we develop the directional derivative that gives the

rate of change of φ at a given r, in the direction of eˆ .

Start with,

dφ = ∂φ dq1 + ∂φ dq2 + ∂φ dq3 = ∂φ dqi .

∂q1 ∂q2 ∂q3 ∂qi

Recall, for a differential displacement,

dr = dqiei.

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Vector Calculus & General Coordinate Systems

Then,

dφ = ∂φ e j ⋅ dqiei

∂q j

= ⎛ ∂φ e1 + ∂φ e2 + ∂φ e3 ⎞ ⋅ dr
⎜⎝ ⎟⎠
∂q1 ∂q2 ∂q3

Now with ds = |dr|, we can define the direction by the unit
vector,
eˆ = dr .

ds

The directional derivative is defined by

⎛ dφ ⎞ = ∂φ ej ⋅ eˆ, (27)
⎝⎜ ⎟⎠eˆ
ds ∂q j 190

Vector Calculus & General Coordinate Systems

and is the rate of change of φ with respect to s in the direction

of eˆ .
Define the vector,

gradφ ≡ ∂φ e j

∂q j (28)

as the gradient vector that points in the direction of maximum

change of φ. Note that grad φ is defined in terms of covariant

components scalar components and the contravariant basis.

For a general representation of level surfaces of φ, i.e., surfaces
for which φ = const, the unit normal vector is the unit vector

locally perpendicular to a level surface,

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Vector Calculus & General Coordinate Systems

gradφ eˆ nˆ

φ = C1 nˆ

φ = C2 dφ = eˆ ⋅ gradφ

r ds

For the surface, the unit normal is (29)

nˆ = ± gradφ . 192
|gradφ|

Vector Calculus & General Coordinate Systems

For a closed surface, it is the convention to choose +nˆ pointing
outward.
Now define the del operator, ∇,

gradφ ≡ ∇φ ⎛ e1 ∂ + e2 ∂ + e3 ∂ ⎞⎠⎟φ ,
⎜ ∂q1 ∂q2 ∂q3
⎝

∇ ≡ ei ∂ . (30)
∂qi

Note that ∇ is a vector differential operator, i.e., it has some

properties of a vector. Since we have defined ∇ to operate

from left to right, it does not share the commutativity property

of a vector for the dot product, i.e.,

a⋅∇ ≠ ∇⋅a

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Vector Calculus & General Coordinate Systems

In fact, the left side of the inequality is itself an operator.
For orthogonal curvilinear coordinates,

ei = eˆi (no summation),
hi

∇ = eˆ1 ∂ + eˆ 2 ∂ + eˆ 3 ∂ . (31)
h1 ∂q1 h2 ∂q2 h3 ∂q3

For the Cartesian system,

∇ = ˆii ∂ . (32)
∂xi (33)

The divergence of a vector a is defined as,

div a ≡ ∇ ⋅a.

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Vector Calculus & General Coordinate Systems

As we will see, ∇ ⋅a is related to the net efflux of some
vector quantity a per unit volume, at a point in space.

In general,

div a = ∂ai δij + ai ⎧k j ⎫⎬δ j
∂q j ⎩⎨i k
⎭

= ∂ai + ai ⎧ i⎫
∂q j ⎨⎩i ⎬.
j ⎭

Now recall another relation you thought we’d never use, eq.
(21),

⎧k k ⎫ = ∂ ln g = 1 ∂g .
⎩⎨i ⎬ ∂qi 2g ∂qi
⎭

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Vector Calculus & General Coordinate Systems

Then,

div a = ∂ai + ai ∂( g )
∂qi g ∂qi .

So we have two alternate forms commonly seen in the
literature,

div a = 1 ∂ ( gai ), (34)
g ∂qi (35)

and

div a = 1 ∂ (Jai ).
J ∂qi

Where J is the Jacobian.

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Vector Calculus & General Coordinate Systems

For orthogonal curvilinear coordinates,
a = aihieˆi = aˆieˆi (aˆi = aihi = aˆi no summation)

Then,

div a = 1 ∂ (h1h2h3ai ), (36)
h1h2h3 ∂qi

= 1 ⎡∂ (h2 h3aˆ1 ) + ∂ (h1h3aˆ2 ) + ∂ (h1h2aˆ3 ) ⎤ . (37)
h1h2h3 ⎢⎣ ∂q1 ∂q2 ∂q3 ⎥⎦

Note we intentionally emphasize the physical components in
the form of eq. (37). And finally, for the Cartesian system,

div a = ∂ai . (39)
∂xi
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Vector Calculus & General Coordinate Systems

Integral Relations (Theorem of Gauss)
The following relations from vector integral calculus relate
surface integrals to volume integrals. As we will see, these
relations can be used to give a more physical interpretation to
some of the vector differential relations defined in the previous
sections. The vector integral relations form the basis of the
integral and differential conservation laws. For development,
we will focus on the conservation laws of continuum
mechanics. For computational methods, they form the basis of
finite-volume and finite-element methods.
We begin with an arbitrary region R in space enclosed by a
surface S,

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Vector Calculus & General Coordinate Systems

dτ n

n dS
S
r
R

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Vector Calculus & General Coordinate Systems

A differential volume element is dτ, and dS is the magnitude of

a differential surface element whose orientation is determined
by the outward unit normal nˆ . Now introduce three vector
integral theorems,

∫∫∫R gradφ dτ = w∫∫S nˆφ dS Gradient Theorem (49)

∫∫∫R diva dτ = w∫∫S nˆ ⋅a dS Divergence Theorem (50)

∫∫∫R curla dτ = w∫∫S nˆ ×a dS Curl Theorem (51)

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Vector Calculus & General Coordinate Systems

Using the divergence theorem, we can create a derivative

relation similar to the directional derivative. Set a = ∇φ,

∫∫∫R div gradφ dτ = ∫∫∫R ∇2φ dτ = w∫∫S nˆ ⋅ gradφ dS (52)

Here, we identify the normal derivative,

nˆ ∇φ ∂φ ≡ nˆ ⋅∇φ (53)

∂n

is the rate of change of

φ in the direction

∂φ normal to the surface S

∂n enclosing region R.

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Vector Calculus & General Coordinate Systems

In Cartesian coordinates, the normal derivative is,

∂φ = niˆii ⋅ ∂φ ˆi j = nj ∂φ (54)

∂n ∂x j ∂x j

Because nˆ is a unit vector, the ni are direction cosines.
Using the integral relations, we can develop more intuitive,
alternative definitions for some of the vector differential
relations introduced earlier. (Note that in the integral approach
we make no references to any coordinate system.)

We begin by examining the limit as the region R shrinks to a
point, i.e.,

R → ∆τ , S → ∆S.

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Vector Calculus & General Coordinate Systems

Why does ∆τ → 0 but S → ∆S ≠ 0? The integral relations are

now,

∫∫∫R gradφ dτ = w∫∫S nˆφ dS → gradφ ∆τ ≅ w∫∫∆S nˆφ dS

∫∫∫R diva dτ = w∫∫S nˆ ⋅a dS → diva ∆τ ≅ w∫∫∆S nˆ ⋅a dS

∫∫∫R curla dτ = w∫∫S nˆ ×a dS → curla ∆τ ≅ w∫∫S nˆ × a dS

Again, since we have not referred to any coordinate system,
and these relations involve vectors, the following are called the
invariant forms,

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Vector Calculus & General Coordinate Systems

w∫∫gradφ ≅ lim 1 nˆ φ dS (55)
τ∆τ →0 ∆
∆S

w∫∫diva ≅ lim 1 nˆ ⋅a dS (56)

τ∆τ →0 ∆ ∆S

w∫∫curla ≅ lim 1 nˆ × a dS (57)

∆τ →0 ∆τ S

Physical interpretations of integral definitions

Note that each of the invariant forms resembles the difference-
quotient definition of a derivative from the differential calculus
(1-D),

du = lim u(x + ∆x) − u(x)
dx ∆x→0 ∆x

where u is some scalar function of x.

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Vector Calculus & General Coordinate Systems

grad φ: φ = C2
φ = C3
φnˆ
φnˆ
S

R φ = C5
φ = C4
φ = C1

In eq. (55) the function φ appears as a weighting function for

nˆ dS , i.e., larger values ofφnˆ dS contribute more to the total

integral than smaller (in magnitude) values. As ∆τ → 0, the

integral is weighted most in the direction of greatest increase

of φ, i.e., the term in the direction of grad φ is the dominant

term in the integral, thus the integral definition corresponds

with the original gradient definition, eq. (28).

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Vector Calculus & General Coordinate Systems

div a: nˆ nˆ v
dS vn
v
S

vt

R

For illustration, set a = v in eq. (56), where v is the velocity of
some quantity flowing from region R through the surface S.
Note that R and S define a bounded region in space, not
necessarily a physical boundary. Then,

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Vector Calculus & General Coordinate Systems

vndS = v ⋅nˆ dS ≡ outflow (efflux) through dS. (58)

As ∆τ → 0, the integral, eq. (56), becomes,

div v ≡ net outflow
unit volume

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Vector Calculus & General Coordinate Systems

curl a:

We use Stokes’ theorem to show the curl is related to the

circulation of a vector field:

curl a

capping

surface nˆ

dS

C ds

a

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Vector Calculus & General Coordinate Systems

∫∫S (curla) ⋅nˆ dS = v∫C a ⋅
ds

circulation of
a about C

Shrink the region until the capping surface just covers a
differential plane area bounded by the curve C, i.e.,

nˆ

|curl a|n curl a C → Cn: bounds
Cn ∆S the infinitesimal

ds plane surface ∆S.

a 209

Vector Calculus & General Coordinate Systems

nˆ ⋅ curla ≅ v∫C a ⋅ds,

Let ∆S → 0,

nˆ ⋅ (curla) = nˆ ⋅ (curla) = lim 1 a ⋅ ds
v∫∆Sn
∆S →0 Cn

nˆ ⋅ (curla) = nˆ ⋅ (curla) ≡ circulation around C .
n area enclosed by C

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Vector Calculus & General Coordinate Systems

Example: Show that in a fluid, the bouyancy force f is equal
and opposite to the local gravitational acceleration g and equal
to the weight of the displaced fluid.

Solution:

Pressure is defined as the inward normal stress on a surface,
i.e., it is the (force)/(unit area) at a point on a surface S,

oriented in direction opposite the outward unit normal nˆ to the

surface. For a body immersed in a static pressure field p(r),
the total force is

f = −w∫∫ pnˆ dS.
S

The magnitude and direction of the greatest increase in

pressure is grad p = ρg, where ρ is the mass density. From the

gradient theorem eq. (49),

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Vector Calculus & General Coordinate Systems

∫∫∫R gradφ dτ = w∫∫S nˆ φdS

Now with the scalar field set to φ(r) = p(r),

∫∫∫R grad p dτ = w∫∫S nˆ p dS

Thus,

f = −w∫∫ pnˆ dS = −∫∫∫R grad p dτ
S
= −∫∫∫R ρg dτ = −g∫∫∫R ρdτ = −mg ⇐ Q.E.D.

Eureka! This is Archimedes’ principal!

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Vector Calculus & General Coordinate Systems

Example: Show that the continuity equation

∂ρ + divρ v = 0

∂t

is the local mass conservation relation. Where t is time, ρ is

density, and v is velocity.
Solution:

Begin with a fixed region in space with, ρ = ρ(r,t), v = v(r,t),

and

ρ v ≡ mass flux= mass .

(area)(time)

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Vector Calculus & General Coordinate Systems
nˆ

ρv

S

r R fixed in space

The mass conservation law is, (with no mass sources)

⎡rate of change of ⎤ = ⎡net flux of mass ⎤
⎣⎢mass in region R ⎦⎥ ⎢⎣across surface S ⎥⎦

d ∫∫∫R ρ dτ = w∫∫S ρ v ⋅nˆ dS
dt

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Vector Calculus & General Coordinate Systems

Since R is fixed in space,

∫∫∫R ∂ρ dτ + w∫∫S ρv ⋅ nˆ dS = 0.

∂t

Applying the divergence theorem, the surface integral is
converted to a volume integral, and

∫∫∫R ⎛ ∂ρ + divρ v ⎞ dτ = 0.
⎜⎝ ⎠⎟
∂t

Since this relation must hold for any arbitrary region R, the
only way the integral is always zero is if the integrand is zero,
thus

∂ρ + divρ v = 0. ⇐ Q.E.D.

∂t

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Vector Calculus & General Coordinate Systems

Reynolds Transport Theorem

The Reynolds transport theorem allows us to develop general
transport relations. For example, in some medium (gas, liquid,
solid, vacuum), we can write relations that describe the
evolution of the field distribution of some vector (tensor) or
scalar quantity due to the transport or convection of medium
quantities (e.g., mass, momentum, energy, etc.)

Recall, in 1-D integro-differential calculus, the derivative of an
integral function I(t) is given by the Liebniz rule,

∫ b(t )

I (t) = f (x)dx,
a(t )

dI = db f (b(t)) − da f (a(t)).
dt dt dt

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Vector Calculus & General Coordinate Systems

For a function of 1-D space and time,

b(t )

∫I (t) = f (x,t)dx,
a(t )

dI = db f (b(t),t) − da f (a(t),t) + b(t) ∂f dx.
∫dt dt dt a(t ) ∂t

Now generalize this to a 3-D system,

nˆ vS Let Q(r,t) represent some
S(t) flow quantity, e.g., density.

In the general case, each

dS point on the surface S(t)
has a velocity vS.

R(t)

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Vector Calculus & General Coordinate Systems

I (t) = ∫∫∫ Q(r,t)dτ the total amount of Q in
region R at time t.

∂Q dτ +

R(t ) ∂t
∫∫∫ w∫∫dI = S (t) Q(r,t)vS ⋅ nˆ dS (59)

dt

For the special case of a material region,
vS = v ≡ velocity of the medium,
i.e., each point on the surface S(t) moves with the local
velocity of the medium. For this case, it is standard to denote
the time derivative with a special notation,

d → D ≡ substantial (material) derivative.
dt Dt

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Vector Calculus & General Coordinate Systems

Then,

∂Q dτ +
∫∫∫ w∫∫DI = Q(r,t)v ⋅nˆ dS (60)
Dt R(t) ∂t
S (t )

Conservation Equations

Using the Reynolds transport theorem, we can develop the
physical conservation laws in the form used in study of
classical tensor, vector, scalar fields.

Mass

The mass dm in a differential volume element dτ is ρdτ. Then

the total mass in the region R(t) is,

∫∫∫ ρdτ.
R(t)

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Vector Calculus & General Coordinate Systems

The statement of mass conservation is that the mass contained
in a region R(t) that is moving at the medium velocity v is

conserved,

∫∫∫D ρ(r,t)dτ = 0. (61)

Dt R(t )

Alternatively, using the Reynolds transport relation eq. (60),

∫∫∫ w∫∫∂ρ dτ + ρ(r,t)v ⋅nˆ dS = 0. (62)
R(t) ∂t S (t )

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Vector Calculus & General Coordinate Systems

Momentum
Newton’s second law is:

⎡sum of external⎤ = ⎡time rate of change⎤
⎣⎢forces ⎥⎦ ⎣⎢of momentum ⎥⎦

∑f = d (mv).
dt

Now write this for a material region,

momentum of mass⎫ → momentum in R(t) is ∫∫∫R (t ) ρ vdτ
⎬
dm is ρ vdτ ⎭

Momentum conservation is then written as

D ∫∫∫R (t ) ρ v(r,t)dτ = ∑f. (63)
Dt
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Vector Calculus & General Coordinate Systems

Alternatively, using the Reynolds transport relation eq. (60),

∫∫∫R (t ) ∂( ρ v) dτ + w∫∫ S (t ) ρ vv ⋅nˆ dS = ∑f. (64)

∂t

Energy
The first law of thermodynamics is

⎡change in system⎤ ⎡heat transfer across⎤ ⎡work on the ⎤
⎢⎣total energy ⎥⎦ ⎢⎣system boundary⎥⎦
= ⎢⎢the boundary into ⎥ +
⎣⎢the system ⎥
⎦⎥

∆E = δ Q + δW

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Vector Calculus & General Coordinate Systems

The total energy in dτ is

⎛ ρ v2 ⎞
⎜ ⎟
total energy in dτ = ⎜⎝⎜ Nρu + N2 ⎟⎠⎟ dτ

potential kinetic

where u is the specific internal energy and v is speed. We then
write the energy equation as

D ∫∫∫R(t ) ρ ⎛ u + v⋅v ⎞ dτ = δQ + δW . (65)
Dt ⎝⎜ 2 ⎠⎟
dt dt

Alternatively, using the Reynolds transport relation eq. (60),

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Vector Calculus & General Coordinate Systems

Alternatively, using the Reynolds transport relation eq. (60),

∂∫∫∫ w∫∫R(t)⎡ρ⎛u + v⋅v ⎞⎤ dτ + S (t ) ρ ⎛ u + v⋅v ⎞ v ⋅nˆ dS = δQ + δW .
∂t ⎣⎢ ⎝⎜ 2 ⎠⎟⎦⎥ ⎝⎜ 2 ⎠⎟
dt dt

(64)

Note that we have explicitly used a ‘δ’ to designate differential

of heat Q and work W. Why?

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