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Copyright Brian J. Kirby. With questions, contact Prof. Kirby here. 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.

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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] [Induced-Charge Effects] [DEP] [Solution Chemistry]

C.2 Vector calculus [coordinates top]

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 microfluidics textbook nanofluidics textbook Brian Kirby Cornell. In this text, we use arrows to distinguish between vectors (e.g., the velocity microfluidics textbook nanofluidics textbook Brian Kirby Cornell) and scalars (e.g., the magnitude of the x-velocity u). We use hats to denote unit vectors, so microfluidics textbook nanofluidics textbook Brian Kirby Cornell , for example, denotes a unit vector in the x-direction. We distinguish between position vectors, distance vectors, velocity vectors, coordinate vectors, unit vectors, and vector magnitudes. For example, microfluidics textbook nanofluidics textbook Brian Kirby Cornell  denotes the position vector relative to the origin, microfluidics textbook nanofluidics textbook Brian Kirby Cornell  denotes a unit vector in that direction, and r denotes the magnitude of microfluidics textbook nanofluidics textbook Brian Kirby Cornell . These are related by microfluidics textbook nanofluidics textbook Brian Kirby Cornell = microfluidics textbook nanofluidics textbook Brian Kirby Cornell ∕r. The distance vector between two points is denoted using microfluidics textbook nanofluidics textbook Brian Kirby Cornell.

Using unit vectors, we can write a vector (for example, the velocity vector microfluidics textbook nanofluidics textbook Brian Kirby Cornell) in Cartesian coordinates as
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(C.6)

in cylindrical coordinates as
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(C.7)

and in spherical coordinates as
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(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:
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(C.9)

or
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(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:

microfluidics textbook nanofluidics textbook Brian Kirby Cornell
and
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
where we use tan-1 to denote the two-argument inverse tangent, whose value (in the range 0-2π) depends on the sign of y and x, as well as the value of y∕x:
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
For spherical coordinates, we have:
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
and
microfluidics textbook nanofluidics textbook Brian Kirby Cornell

microfluidics textbook nanofluidics textbook Brian Kirby Cornell


Magnitude of a velocity vector The magnitude of a velocity vector microfluidics textbook nanofluidics textbook Brian Kirby Cornell is denoted in this text with absolute value symbols: |microfluidics textbook nanofluidics textbook Brian Kirby Cornell|. 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 microfluidics textbook nanofluidics textbook Brian Kirby Cornell = microfluidics textbook nanofluidics textbook Brian Kirby Cornellu+ŷv+w, the magnitude is given by the Euclidean norm:
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(C.25)

while the magnitude of a cylindrical velocity vector microfluidics textbook nanofluidics textbook Brian Kirby Cornell = microfluidics textbook nanofluidics textbook Brian Kirby Cornellur+microfluidics textbook nanofluidics textbook Brian Kirby Cornelluθ+uz is given by
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(C.26)

and the magnitude of a spherical velocity vector microfluidics textbook nanofluidics textbook Brian Kirby Cornell = microfluidics textbook nanofluidics textbook Brian Kirby Cornell ur+microfluidics textbook nanofluidics textbook Brian Kirby Cornelluϑ+microfluidics textbook nanofluidics textbook Brian Kirby Cornelluφ is given by
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(C.27)

Length of position vector The length of a position vector microfluidics textbook nanofluidics textbook Brian Kirby Cornell  is also denoted in this text with absolute value symbols: |microfluidics textbook nanofluidics textbook Brian Kirby Cornell |. 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 microfluidics textbook nanofluidics textbook Brian Kirby Cornell = (x, y, z), the magnitude is given by the Euclidean norm:
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(C.28)

while the magnitude of a cylindrical position vector microfluidics textbook nanofluidics textbook Brian Kirby Cornell  specified by r, θ, and z is given by
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(C.29)

and the magnitude of a spherical position vector microfluidics textbook nanofluidics textbook Brian Kirby Cornell  specified by r, ϑ, and φ is given by
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(C.30)

Length of distance vector The distance vector microfluidics textbook nanofluidics textbook Brian Kirby Cornell from a point denoted by position vector microfluidics textbook nanofluidics textbook Brian Kirby Cornell 1 to a point denoted by position vector microfluidics textbook nanofluidics textbook Brian Kirby Cornell 2 is written as microfluidics textbook nanofluidics textbook Brian Kirby Cornell = microfluidics textbook nanofluidics textbook Brian Kirby Cornell 2 -microfluidics textbook nanofluidics textbook Brian Kirby Cornell 1. In Cartesian coordinates, the value of the components of this vector are simply the difference between the components of the position vectors:
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(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.


microfluidics textbook nanofluidics textbook Brian Kirby Cornell


Generally, an nth-order tensor or a rank n tensor refers to a class of properties that have magnitudes with n directional components. So a scalar is a zeroth-order tensor (no direction) and a vector is a first-ordertensor (one direction). In this text, however, we use the term tensor to refer specifically to second-order tensors, i.e., values that have magnitude and two directions. For example, the velocity gradient microfluidics textbook nanofluidics textbook Brian Kirby Cornell 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., microfluidics textbook nanofluidics textbook Brian Kirby Cornell, the strain rate tensor ), vectors (e.g., microfluidics textbook nanofluidics textbook Brian Kirby Cornell, the velocity), and scalars (e.g., u, the magnitude of the x-velocity). 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
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(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):
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(C.33)

where α is the angle between vectors microfluidics textbook nanofluidics textbook Brian Kirby Cornell and microfluidics textbook nanofluidics textbook Brian Kirby Cornell. The dot product commutes, so microfluidics textbook nanofluidics textbook Brian Kirby Cornellmicrofluidics textbook nanofluidics textbook Brian Kirby Cornell = microfluidics textbook nanofluidics textbook Brian Kirby Cornellmicrofluidics textbook nanofluidics textbook Brian Kirby Cornell.

For position vectors, the fact that the coordinates have different units for different coordinate systems makes the definition of the dot product coordinate system-specific. For two Cartesian position vectors microfluidics textbook nanofluidics textbook Brian Kirby Cornell = (x1,y1,z1) and microfluidics textbook nanofluidics textbook Brian Kirby Cornell = (x2,y2,z2), the dot product is given by
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(C.34)

but for two cylindrical position vectors microfluidics textbook nanofluidics textbook Brian Kirby Cornell = (r11,z1) and microfluidics textbook nanofluidics textbook Brian Kirby Cornell = (r22,z2), the dot product is given by
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(C.35)

and for two spherical position vectors microfluidics textbook nanofluidics textbook Brian Kirby Cornell = (r1,ϑ11) and microfluidics textbook nanofluidics textbook Brian Kirby Cornell = (r2,ϑ22), the dot product is given by
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(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 ??.


microfluidics textbook nanofluidics textbook Brian Kirby CornellFigure C.2: Definition of α for the calculation of the dot product.



microfluidics textbook nanofluidics textbook Brian Kirby Cornell


Matrix representation for Cartesian vectors and tensors For two Cartesian vectors microfluidics textbook nanofluidics textbook Brian Kirby Cornell = (x1, y1, z1) and microfluidics textbook nanofluidics textbook Brian Kirby Cornell = (x2, y2, z2), we can write the dot product as:

microfluidics textbook nanofluidics textbook Brian Kirby Cornell

and the dot product for a Cartesian vector microfluidics textbook nanofluidics textbook Brian Kirby Cornell = (u, v, w) and a Cartesian tensor τ can be written as:

microfluidics textbook nanofluidics textbook Brian Kirby Cornell

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):
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(C.39)

where α is the angle between the two vectors and microfluidics textbook nanofluidics textbook Brian Kirby Cornell is a unit normal to the plane of the two vectors microfluidics textbook nanofluidics textbook Brian Kirby Cornell and microfluidics textbook nanofluidics textbook Brian Kirby Cornell, with a direction given by the right-hand rule. Note that the cross product anticommutes, i.e., microfluidics textbook nanofluidics textbook Brian Kirby Cornell×microfluidics textbook nanofluidics textbook Brian Kirby Cornell = -microfluidics textbook nanofluidics textbook Brian Kirby Cornell×microfluidics textbook nanofluidics textbook Brian Kirby Cornell. 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 system-specific. For two Cartesian position vectors microfluidics textbook nanofluidics textbook Brian Kirby Cornell = (x1,y1,z1) and microfluidics textbook nanofluidics textbook Brian Kirby Cornell = (x2,y2,z2), the cross product is given by
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(C.40)

For cylindrical and spherical coordinates, we convert to Cartesian coordinates to evaluate the cross product.


microfluidics textbook nanofluidics textbook Brian Kirby CornellFigure C.3: Definition of α and microfluidics textbook nanofluidics textbook Brian Kirby Cornell for the calculation of the cross product.


Matrix representation for Cartesian vectors For Cartesian tensors, we can write the cross product for two vectors:

microfluidics textbook nanofluidics textbook Brian Kirby Cornell

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 three-dimensional 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 2nd-order 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
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(C.42)

Matrix representation of Cartesian gradients In Cartesian coordinates, the gradient of a scalar a is given by

microfluidics textbook nanofluidics textbook Brian Kirby Cornell

and the gradient of a vector (u,v,w) is given by

microfluidics textbook nanofluidics textbook Brian Kirby Cornell

Note that the gradient of a scalar results in a vector, and the gradient of a vector results in a 2nd-order 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 , we move a distance r. 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
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(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 , we move a distance rdϑ, and if we rotate by an angle , 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
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(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 microfluidics textbook nanofluidics textbook Brian Kirby Cornell is defined as
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(C.47)

where ΔV is a volume, S is its surface, and dA is a differential area element along that surface. microfluidics textbook nanofluidics textbook Brian Kirby Cornell 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:

microfluidics textbook nanofluidics textbook Brian Kirby Cornell

Divergence Operator–Cartesian Coordinates InCartesian coordinates, the divergence of the velocity vector microfluidics textbook nanofluidics textbook Brian Kirby Cornell is given by

microfluidics textbook nanofluidics textbook Brian Kirby Cornell

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 microfluidics textbook nanofluidics textbook Brian Kirby Cornell term must be corrected by a factor of r. Also, the microfluidics textbook nanofluidics textbook Brian Kirby Cornell term must account for the fact that the geometry changes with r. Thus, the divergence of the velocity vector microfluidics textbook nanofluidics textbook Brian Kirby Cornell in cylindrical coordinates is given by

microfluidics textbook nanofluidics textbook Brian Kirby Cornell

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 microfluidics textbook nanofluidics textbook Brian Kirby Cornell in cylindrical coordinates is given by

microfluidics textbook nanofluidics textbook Brian Kirby Cornell

Curl operator

The curl (∇×) of the velocity vector tells to what extent the fluid is rotating. Formally, the curl of a vector microfluidics textbook nanofluidics textbook Brian Kirby Cornell is defined as
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(C.52)

where ΔS is an area, C  is its surface, and ds is a differential element along that contour. microfluidics textbook nanofluidics textbook Brian Kirby Cornell 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:

microfluidics textbook nanofluidics textbook Brian Kirby Cornell

Curl Operator–Cartesian Coordinates The curl ∇×microfluidics textbook nanofluidics textbook Brian Kirby Cornell of a vector microfluidics textbook nanofluidics textbook Brian Kirby Cornell is given, in Cartesian coordinates, as

microfluidics textbook nanofluidics textbook Brian Kirby Cornell

Matrix representation of curl for Cartesian coordinates The curl in Cartesian coordinatescan also be written as

microfluidics textbook nanofluidics textbook Brian Kirby Cornell

Curl Operator–Cylindrical Coordinates The curl∇×microfluidics textbook nanofluidics textbook Brian Kirby Cornell of a vector microfluidics textbook nanofluidics textbook Brian Kirby Cornell is given, in cylindrical coordinates, as

microfluidics textbook nanofluidics textbook Brian Kirby Cornell

Curl Operator–Spherical Coordinates Thecurl ∇×microfluidics textbook nanofluidics textbook Brian Kirby Cornell of a vector microfluidics textbook nanofluidics textbook Brian Kirby Cornell is given, in spherical coordinates, as

microfluidics textbook nanofluidics textbook Brian Kirby Cornell

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 microfluidics textbook nanofluidics textbook Brian Kirby Cornell is defined as
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(C.58)

The Laplacian (2) of the velocity vector is a vector quantity, and is computed in the same way:
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(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:
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(C.60)

In Cartesian coordinates, the components of the Laplacian 2microfluidics textbook nanofluidics textbook Brian Kirby Cornell of a vector microfluidics textbook nanofluidics textbook Brian Kirby Cornell are given by the Laplacians of the vector components:

microfluidics textbook nanofluidics textbook Brian Kirby Cornell

Laplacian Operator–Cylindrical Coordinates TheLaplacian of a scalar in cylindrical coordinates is given by
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(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 ∇×microfluidics textbook nanofluidics textbook Brian Kirby Cornell of a vector microfluidics textbook nanofluidics textbook Brian Kirby Cornell in cylindrical coordinates, is given by

microfluidics textbook nanofluidics textbook Brian Kirby Cornell

Laplacian Operator–Spherical Coordinates The Laplacian 2ϕ of a scalar ϕ is given, in spherical coordinates, as

microfluidics textbook nanofluidics textbook Brian Kirby Cornell

The Laplacian of a vector microfluidics textbook nanofluidics textbook Brian Kirby Cornell is given in spherical coordinates by

microfluidics textbook nanofluidics textbook Brian Kirby Cornell

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:
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(C.75)

The divergence of the curl of a vector field is always zero:
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(C.76)

Laplacian
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(C.77)

microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(C.78)

Product rules Product rule for the gradient:
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(C.79)

Vector operations on the product of a scalar and a vector:
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(C.80)

microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(C.81)

Gradient of vector dot product:
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(C.82)

Divergence of vector cross product:
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(C.83)

Curl of vector cross product:
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(C.84)

Commutations of vector operations with the Laplacian The cross product, dot product, and gradient all commute with the Laplacian:
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(C.85)

microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(C.86)

microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(C.87)


microfluidics textbook nanofluidics textbook Brian Kirby Cornell


C.2.5 Dyadic operations

The superposition of two vectors (for example, microfluidics textbook nanofluidics textbook Brian Kirby Cornellmicrofluidics textbook nanofluidics textbook Brian Kirby Cornell) denotes a second rank dyadictensor, and the dot product of a dyadic tensor with a vector results in a vector.
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(C.93)

The dyadic tensor microfluidics textbook nanofluidics textbook Brian Kirby Cornellmicrofluidics textbook nanofluidics textbook Brian Kirby Cornell, when multiplying by a vector microfluidics textbook nanofluidics textbook Brian Kirby Cornell, gives a vector that is in the direction of microfluidics textbook nanofluidics textbook Brian Kirby Cornell with a magnitude given by the dot product microfluidics textbook nanofluidics textbook Brian Kirby Cornellmicrofluidics textbook nanofluidics textbook Brian Kirby Cornell. 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:
microfluidics textbook nanofluidics textbook Brian Kirby Cornell
(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] [Induced-Charge 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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