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[Return to Table of Contents]
Jump To:
[Kinematics]
[Couette/Poiseuille Flow]
[Fluid Circuits]
[Mixing]
[Electrodynamics]
[Electroosmosis]
[Potential Flow]
[Stokes Flow]
[Debye Layer]
[Zeta Potential]
[Species Transport]
[Separations]
[Particle Electrophoresis]
[DNA]
[Nanofluidics]
[InducedCharge Effects]
[DEP]
[Solution Chemistry]
Spatial phenomena naturally give rise to vector descriptions, and the calculus of these parameters is central to all of
the analysis in this text. This section describes scalars, vectors, and tensors, as well as calculus operations on these
quantities.
C.2.1 Scalars, vectors, and tensors
We use scalars, vectors, and tensors to denote a variety of properties. Scalars indicatevalues that have a magnitude
but no direction, for example pressure p and temperature T . Vectors indicatevalues that have both magnitude and
direction, for example the velocity . In this text, we use arrows to distinguish between vectors (e.g., the velocity
) and scalars (e.g., the magnitude of the xvelocity u). We use hats to denote unit vectors, so , for
example, denotes a unit vector in the xdirection. We distinguish between position vectors, distance
vectors, velocity vectors, coordinate vectors, unit vectors, and vector magnitudes. For example, denotes
the position vector relative to the origin, denotes a unit vector in that direction, and r denotes the
magnitude of . These are related by = ∕r. The distance vector between two points is denoted using
.
Using unit vectors, we can write a vector (for example, the velocity vector ) in Cartesian coordinates
as
 (C.6) 
in cylindrical coordinates as
 (C.7) 
and in spherical coordinates as
 (C.8) 
The components of a Cartesian vector can be written as a matrix. This matrix notation is convenient and useful
for Cartesian vectors, because matrix operations can determine the result of a vector operation. However, the matrix
notation is not well suited for cylindrical or spherical coordinate systems, since the vector operations for curvilinear
systems do not correspond to simple matrix operations. Because of this, we specifically avoid matrix representations
for curvilinear coordinate systems.
In Cartesian coordinates, we find it useful to write vectors as a 3×1 or 1×3 matrix, for example:
 (C.9) 
or
 (C.10) 
Conversions between coordinate systems
We often find it useful to convert back and forth between coordinate systems. Most often, we convert back and
forth between Cartesian coordinates and other coordinate systems. For cylindrical coordinates, we have:
and where we use tan^{1} to denote the twoargument inverse tangent, whose value (in the range 02π) depends on the
sign of y and x, as well as the value of y∕x: For spherical coordinates, we have: and
Magnitude of a velocity vector
The magnitude of a velocity vector is denoted in this text with absolute value symbols: . Since each
component of velocity is orthogonal to all other components and has the same units, the expression for
the magnitude of a velocity vector is essentially the same in any of the three coordinate systems we
typically use. For Cartesian velocity vectors = u+ŷv+ẑw, the magnitude is given by the Euclidean
norm:
 (C.25) 
while the magnitude of a cylindrical velocity vector = u_{r}+u_{θ}+ẑu_{z} is given by
 (C.26) 
and the magnitude of a spherical velocity vector = u_{r}+u_{ϑ}+u_{φ} is given by
 (C.27) 
Length of position vector
The length of a position vector is also denoted in this text with absolute value symbols:  . As compared to
velocity vectors, position vectors are less straightforward to write. The reason for this is the conventions used
for writing these vectors. The coordinates x, y, z, r, and r have units of length, while θ, ϑ, and φ are
unitless.
For Cartesian position vectors = (x, y, z), the magnitude is given by the Euclidean norm:
 (C.28) 
while the magnitude of a cylindrical position vector specified by r, θ, and z is given by
 (C.29) 
and the magnitude of a spherical position vector specified by r, ϑ, and φ is given by
 (C.30) 
Length of distance vector
The distance vector from a point denoted by position vector _{1} to a point denoted by position vector _{2} is
written as = _{2}  _{1}. In Cartesian coordinates, the value of the components of this vector are simply the difference
between the components of the position vectors:
 (C.31) 
and the length of this vector can be calculated by evaluating the Euclidean norm. In cylindrical and spherical
coordinates, the length of the distance vector is usually calculated by first converting to Cartesian coordinates.
Generally, an nthorder tensor or a rank n tensor refers to a class of properties that have magnitudes with n
directional components. So a scalar is a zerothorder tensor (no direction) and a vector is a firstordertensor (one
direction). In this text, however, we use the term tensor to refer specifically to secondorder tensors, i.e., values that
have magnitude and two directions. For example, the velocity gradient ∇ has values that are a function of two
directions: the direction of the velocity component and the direction of its gradient. In this text, we
use arrows or double arrows to distinguish between tensors (e.g., , the strain rate tensor ), vectors
(e.g., , the velocity), and scalars (e.g., u, the magnitude of the xvelocity). Bold type also highlight the
presence of vector or tensor. The use of bold type and arrows simultaneously is a bit redundant, but
combines a notation that is clear in typeset with one that translates well to a chalkboard or to handwritten
notes.
For Cartesian coordinates, the components of a tensor can usefully be written as a matrix. We typically write
Cartesian tensors as 3×3 matrices, for example
 (C.32) 
C.2.2 Vector operations
Key vector operations include thedot product and thecross product. These operations are the vector analogs of scalar
multiplication, providing results that are proportional to the products of the magnitudes of the vectors, but with
additional information related to the direction or orientation of the vectors with respect to each other and/or the
coordinate system.
Dot product
The dot product or scalar productof two vectors is a scalar equal to the product of the length of the two vectors and
the cosine of the subtended angle (Figure C.2):
 (C.33) 
where α is the angle between vectors and . The dot product commutes, so ⋅ = ⋅.
For position vectors, the fact that the coordinates have different units for different coordinate systems makes the
definition of the dot product coordinate systemspecific. For two Cartesian position vectors = (x_{1},y_{1},z_{1}) and
= (x_{2},y_{2},z_{2}), the dot product is given by
 (C.34) 
but for two cylindrical position vectors = (r_{1},θ_{1},z_{1}) and = (r_{2},θ_{2},z_{2}), the dot product is given
by
 (C.35) 
and for two spherical position vectors = (r_{1},ϑ_{1},φ_{1}) and = (r_{2},ϑ_{2},φ_{2}), the dot product is given
by
 (C.36) 
The dot product is used in several ways. One important use is to determine what component of a vector is pointing in
a specific direction. For example, we often evaluate fluxes across surfaces of control volumes; when we do this we
invariably calculate the dot product of a vector with the unit outward normal. Some examples of this implementation
are Equations 5.4, ??, and ??.
Matrix representation for Cartesian vectors and tensors
For two Cartesian vectors = (x_{1}, y_{1}, z_{1}) and = (x_{2}, y_{2}, z_{2}), we can write the dot product as:
and the dot product for a Cartesian vector = (u, v, w) and a Cartesian tensor τ can be written as:
For Cartesian vectors and tensors, these operations are reminiscent of matrix operations, and writing Cartesian
vectors and tensors in matrix form makes these operations straightforward. This approach works only for
rectangular coordinate systems, and curvilinear coordinate system are not generally described using matrix
operations.
Note that, independent of the coordinate system, the dot product of a vector with a vector results in a scalar, and
the dot product of a vector with a tensor results in a vector.
Cross product
The cross productor vector product of two vectors is a vector whose magnitude is equal to the product of the length
of the two vectors and the sine of the subtended angle, and whose direction is normal to plane of the two vectors
(Figure C.3):
 (C.39) 
where α is the angle between the two vectors and is a unit normal to the plane of the two vectors and , with a
direction given by the righthand rule. Note that the cross product anticommutes, i.e., × = ×. Cross
products and results of related vector operations (such as torque or vorticity or angular momentum) are most
precisely termed pseudovectors to denote their behavior upon certain coordinate transformations, but for simplicity
we refer to these quantities as vectors.
For position vectors, the fact that the coordinates have different units for different coordinate systems makes the
definition of the cross product coordinate systemspecific. For two Cartesian position vectors = (x_{1},y_{1},z_{1}) and
= (x_{2},y_{2},z_{2}), the cross product is given by
 (C.40) 
For cylindrical and spherical coordinates, we convert to Cartesian coordinates to evaluate the cross
product.
Matrix representation for Cartesian vectors
For Cartesian tensors, we can write the cross product for two vectors:
Note that the cross product of a vector with a vector results in a vector that is orthogonal to both of the original
vectors.
C.2.3 Del or nabla operations
The symbol ∇is referred to as the del operator or nabla operator. The del operator is avector operator. It is not a
vector. Vector operators operate on scalars, vectors, or tensors to give a result, which may be a scalar or vector or
tensor, depending on the operation.
We use three different vector operations that are denoted using the symbol ∇: thegradient (∇),divergence (∇⋅),
andcurl (∇×) operations. The divergence and curl symbols are reminiscent of the dot and cross product operations,
as is their implementation, especially in Cartesian coordinates. The gradient is a way of taking derivatives that gives
the direction of the maximum rate of change as well as the magnitude of that rate of change. The divergence of a
velocity field tells us how much the flow is expanding or contracting. The curl of a velocity field tells us how much
the flow is rotating.
Gradient operator
The gradient (∇) is essentially a threedimensional spatial derivative. It gives the direction of the maximum rate of
change as well as the magnitude of that rate of change.
The gradient of a scalar is a vector pointing in the direction in which the partial derivative of the scalar in that
direction is maximum, and has a magnitude equal to the spatial derivative with respect to that direction. The gradient
of a vector is a 2ndorder tensor which, in Cartesian coordinates, is made up of vectors corresponding to the
gradients of the components of the vector.
Gradient Operator–Cartesian Coordinates
In Cartesian coordinates, the gradient of a scalar ϕ is given by
 (C.42) 
Matrix representation of Cartesian gradients
In Cartesian coordinates, the gradient of a scalar a is given by
and the gradient of a vector (u,v,w) is given by
Note that the gradient of a scalar results in a vector, and the gradient of a vector results in a 2ndorder
tensor. The units of the gradient are equal to the units of the scalar/vector operated upon, divided by
length.
Gradient Operator–Cylindrical Coordinates
Cartesiancoordinates are relatively straightforward, since x, y, and z are in the same units and all of the relations
in the previous section had a convenient symmetry to them. In cylindrical coordinates, however, r and z are in the
same units, but θ is an angle rather than a length. Specifically, at a given radius r, if we rotate by an angle dθ, we
move a distance rdθ. Thus derivatives with respect to θ differ from derivatives with respect to r by a
factor of r. To account for this, the definition of the gradient operator in cylindrical coordinates is
different than that for Cartesian coordinates. In cylindrical coordinates, the gradient of a scalar ϕ is given
by
 (C.45) 
Gradient Operator–Spherical Coordinates
In spherical coordinates, r has units of length, but ϑ and φ are angles. Specifically, at a given radius r, if we
rotate by an angle dϑ, we move a distance rdϑ, and if we rotate by an angle dφ, we move a distance rsinϑdφ. Thus
derivatives with respect to φ differ from derivatives with respect to r by a factor of rsinϑ. To account for this,
the definition of the gradient operator in spherical coordinates takes a different form than it does in
Cartesian or cylindrical coordinates. So, the gradient of a scalar ϕ in spherical coordinates is given
by
 (C.46) 
Divergence operator
Given a vector that measures the flux density of a property, the divergence (∇⋅) of a the vector measures whether that
property is being created or destroyed. So, the divergence of the velocity vector (which measures volumetric flux
density) measures whether volume is being created or destroyed. For incompressible systems, this is
equivalent to measuring if mass is created or destroyed. In incompressible systems, conservation of
mass implies that the divergence of velocity is zero. Formally, the divergence of a vector is defined
as
 (C.47) 
where ΔV is a volume, S is its surface, and dA is a differential area element along that surface. is a unit outward
normal vector along the surface. This definition naturally leads to thedivergence theorem, which relates the integral
of the flux of a vector through a surface to the volume integral of the divergence:
Divergence Operator–Cartesian Coordinates
InCartesian coordinates, the divergence of the velocity vector is given by
Divergence Operator–Cylindrical Coordinates
Aswas the case with the gradient operator, the fact that cylindrical coordinates have different units makes the
divergence more complicated. As before, the term must be corrected by a factor of r. Also, the term must
account for the fact that the geometry changes with r. Thus, the divergence of the velocity vector in cylindrical
coordinates is given by
Divergence Operator–Spherical Coordinates
As for the cylindrical case, the derivative terms in the divergence operator must account for the fact that the
geometry changes with r and θ. Thus, the divergence of the velocity vector in cylindrical coordinates is given
by
Curl operator
The curl (∇×) of the velocity vector tells to what extent the fluid is rotating. Formally, the curl of a vector is
defined as
 (C.52) 
where ΔS is an area, C is its surface, and ds is a differential element along that contour. is a unit outward normal
along the surface. This definition naturally leads toStokes’ theorem, which relates the integral of the flux of the curl
of a vector across a surface to the contour integral of the vector:
Curl Operator–Cartesian Coordinates
The curl ∇× of a vector is given, in Cartesian coordinates, as
Matrix representation of curl for Cartesian coordinates
The curl in Cartesian coordinatescan also be written as
Curl Operator–Cylindrical Coordinates
The curl∇× of a vector is given, in cylindrical coordinates, as
Curl Operator–Spherical Coordinates
Thecurl ∇× of a vector is given, in spherical coordinates, as
Laplacian operator
The Laplacian(∇^{2}) of a scalar is a scalar that is one of several measures of the curvature of the scalar’s distribution in
space. Formally, the Laplacian of a vector is defined as
 (C.58) 
The Laplacian (∇^{2}) of the velocity vector is a vector quantity, and is computed in the same way:
 (C.59) 
Laplacian Operator–Cartesian Coordinates
The Laplacian of a scalar in Cartesian coordinates is the sum of the second derivatives of the scalar with respect
to the coordinates:
 (C.60) 
In Cartesian coordinates, the components of the Laplacian ∇^{2} of a vector are given by the Laplacians of the
vector components:
Laplacian Operator–Cylindrical Coordinates
TheLaplacian of a scalar in cylindrical coordinates is given by
 (C.64) 
Unfortunately, the Laplacian of a vector in curvilinear coordinates is not given by the the sum of the Laplacians of
the components of the vector. Rather, the Laplacian ∇× of a vector in cylindrical coordinates, is given by
Laplacian Operator–Spherical Coordinates
The Laplacian ∇^{2}ϕ of a scalar ϕ is given, in spherical coordinates, as
The Laplacian of a vector is given in spherical coordinates by
C.2.4 Vector identities
A list of some common vector identities is below.
Null operator combinations
The curl of the gradient of a scalar field is always zero:
 (C.75) 
The divergence of the curl of a vector field is always zero:
 (C.76) 
Laplacian
 (C.77) 
 (C.78) 
Product rules
Product rule for the gradient:
 (C.79) 
Vector operations on the product of a scalar and a vector:
 (C.80) 
 (C.81) 
Gradient of vector dot product:
 (C.82) 
Divergence of vector cross product:
 (C.83) 
Curl of vector cross product:
 (C.84) 
Commutations of vector operations with the Laplacian
The cross product, dot product, and gradient all commute with the Laplacian:
 (C.85) 
 (C.86) 
 (C.87) 
C.2.5 Dyadic operations
The superposition of two vectors (for example, ) denotes a second rank dyadictensor, and the dot product of a
dyadic tensor with a vector results in a vector.
 (C.93) 
The dyadic tensor , when multiplying by a vector , gives a vector that is in the direction of with a magnitude
given by the dot product ⋅. Unlike the dot product of vectors, the dot product of a dyadic with a vector is not in
general equal to the dot product of the vector with the dyadic:
 (C.94) 
[Return to Table of Contents]
Jump To:
[Kinematics]
[Couette/Poiseuille Flow]
[Fluid Circuits]
[Mixing]
[Electrodynamics]
[Electroosmosis]
[Potential Flow]
[Stokes Flow]
[Debye Layer]
[Zeta Potential]
[Species Transport]
[Separations]
[Particle Electrophoresis]
[DNA]
[Nanofluidics]
[InducedCharge Effects]
[DEP]
[Solution Chemistry]
Copyright Brian J. Kirby. Please contact Prof. Kirby here with questions or corrections.
This material may not be distributed without the author's consent. When linking to these pages, please use the URL http://www.kirbyresearch.com/textbook.
This web posting is a draft, abridged version of the Cambridge University Press text. Follow the links to buy at Cambridge or Amazon or Powell's or Barnes and Noble. Contact Prof. Kirby
here. Click here for the most recent version of the errata for the print version.
Ad revenue from these pages is used to support student research. The presence of an advertisement on these pages does not
constitute an endorsement by the Kirby Research Group or Cornell University.
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