The trajectory of a particle moving is space defines a curve that can be defined with time as parameter along the curve. A curve in space is also defined by the intersection of two surfaces, but points along the curve are not associated with time. We will show that a natural parameter for both curves is the distance along the curve.

The variable position vector x(t) describes the motion of a particle. For a finite interval of t, say a≤t≤b, we can plot the position as a curve in space. If the curve does not cross itself (i.e., if ) it is called simple; if x(a)=x(b) the curve is closed. The variable t is now just a parameter along the curve that may be thought of as the time in motion of the particle. If t and t^' are the parameters of two points, the cord joining them is the vector . As t→t^' this vector approaches and so in the limit is proportional to . However the limit of the cord is the tangent so that is in the direction of the tangent. If we can construct a unit tangent vector τ.

(3.3)

Now we will parameterize a curve with distance along the curve rather than time. If x(t) and x(t+dt) are two very close points,

(3.4)

and the distance between them is

(3.5)

The arc length from any given point t=a is therefore

(3.6)

s is the natural parameter to use on the curve, and we observe that

(3.7)

A curve for which a length can be so calculated is called rectifiable. From this point on we will regard s as the parameter, identifying t with s and letting the dot denote differentiation with respect to s. Thus

(3.8)

is the unit tangent vector. Let x(s),x(s+ds), and x(s–ds) be three nearby points on the curve. A plane that passes through these three points is defined by the linear combinations of the cord vectors joining the points. This plane containing the points must also contain the vectors

(3.9)

Thus, in the limit when the points are coincident, the plane reaches a limiting position defined by the first two derivatives and . This limiting plane is called the osculating plane and the curve appears to lie in this plane in the intermediate neighborhood of the point. To prove this statement: (1) A plane is defined by the two vectors, and , if they are not co-linear. (2) The coordinates of the three points on the curve in the previous two equations are a linear combination of and , thus they line in the plane.

Now so and since τ•τ=1,

(3.10)

so that the vector is at right angles to the tangent. Let 1/ρ denote the magnitude of .

(3.11)

Then ν is a unit normal and defines the direction of the so-called principle normal to the curve.

To interpret ρ , we observe that the small angle dθ between the tangents at s and s+ds is given by

(3.12)

since and so . Thus,

(3.13)

is the reciprocal of the rate of change of the angle of the tangent with arc length, i.e., ρ is the radius of curvature. Its reciprocal 1/ρ is the curvature, κ≡|dθ/ds|=1/ρ.

A second normal to the curve may be taken to form a right-hand system with τ and ν . This is called the unit binormal,

If F(x) is a function of position and C is a curve composed of connected arcs of simple curves, x=x(t),a≤t≤b or x=x(s),a≤s≤b, we can define the integral of F along C as

(3.15)

Henceforth, we will assume that the curve has been parameterized with respect to distance along the curve, s.

The integral is from a to b. If the integral is in the opposite direction with opposite limits, then the integral will have the same magnitude but opposite sign. If x(a)=x(b), the curve C is closed and the integral is sometimes written

If the integral around any simple closed curve vanishes, then the value of the integral from any pair of points a and b is independent of path. To see this we take any two paths between a and b, say C₁ and C₂, and denote by C the closed path formed by following C₁ from a to b and C₂ back from b to a.

(3.17)

If a(x) is any vector function of position, a•τ is the projection of a tangent to the curve. The integral of a•τ around a simple closed curve C is called the circulation of a around C.

(3.18)

We will show later that a vector whose circulation around any simple closed curve vanishes is the gradient of a scalar.

Many types of surfaces and considered in transport phenomena. Most often the surfaces are the boundaries of volumetric region of space where boundary conditions are specified. The surfaces could also be internal boundaries where the material properties change between two media. Finally the surface itself may the subject of interest, e.g. the statics and dynamics of soap films.

A proper mathematical treatment of surfaces requires some definitions. A closed surface is one which lies within a bounded region of space and has an inside and outside. If the normal to the surface varies continuously over a part of the surface, that part is called smooth. The surface may be made up of a number of subregions, which are smooth and are called piece-wise smooth. A closed curve on a surface, which can be continuously shrunk to a point, is called reducible. If all closed curves on a surface are reducible, the surface is called simply connected. The sphere is simply connected but a torus is not.

If a surface is not closed, it normally has a space curve as its boundary, as for example a hemisphere with the equator as boundary. It has two sides if it is impossible to go from a point on one side to the other along a continuous curve that does not cross the boundary curve. The surface is sometimes called the cap of the space curve.

If S is a piece-wise smooth surface with two sides in three-dimensional space, we can divide it up into a large number of small surface regions such that the dimensions of the regions go to zero as the number of regions go to infinity. If the regions fill the surface and are not overlapping, then sum of the areas of the regions is equal to the area of the surface. If the function, F is defined on the surface, it can be evaluated for some point of each subregion of the surface and the sum ΣFΔS computed. The limit as the number of regions go to infinity and the dimensions of the regions go to zero is surface integral of F over S.

(3.19)

The traditional symbol of the double integral is retained because if the surface is a plane or the surface is projected on to a plane, then Cartesian coordinates can be defined such that the surface integral is a double integral of the two coordinates in the plane. Also, two surface coordinates can define a surface and the double integration is over the surface integrals.

In transport phenomena the surface integral usually represents the flow or flux of a quantity across the surface and the function F is the normal component of a vector or the contracted product of a tensor with the unit normal vector. Thus one needs to know the direction of the normal in addition to the differential area to calculate the surface integral. Consider the case of a surface defined as a function of two surface coordinates.

(3.20)

To see how we arrive at this result, recall the partial derivatives of the coordinates of a curve with respect to a parameter is a vector that is tangent to the curve. The magnitude is

(3.21)

The vector product has a magnitude equal to the product of the magnitudes and the sine of the angle between the vectors. This gives us the area of a parallelogram corresponding to the area of the differential region.

(3.22)

The two tangent vectors in the direction of the surface coordinates lie in the tangent plane of the surface. Thus the direction of the vector product is perpendicular to the surface. Inward or outward direction for the normal has not yet been specified and will be determined by the sign.

The volume integral of a function F over a volumetric region of space V is the limit of the sum of the products of the volume of small volumetric subregions of V and the function F evaluated somewhere within each subregion.

(3.23)

In Cartesian coordinates the elements of volume dV is simply the volume of a rectangular parallelepiped of sides dx₁, dx₂, dx₃ and so

(3.24)

Suppose, however, that it is convenient to describe the position by some other coordinates, say ξ₁, ξ₂, ξ₃. We may ask what volume is to be associated with the three small changes dξ₁, dξ₂, dξ₃.

The change of coordinates must be given by specifying the Cartesian point x that is to correspond to a given set ξ₁, ξ₂, ξ₃, by

(3.25)

Then by partial differentiation the small differences corresponding to a change dξ_i are

(3.26)

Let d<