Website owner:  James Miller

     VECTOR INTEGRATION, LINE INTEGRALS, SURFACE INTEGRALS, VOLUME INTEGRALS

Integration of vector functions. In ordinary calculus we compute integrals of real functions of a real variable; that is, we compute integrals of functions of the type

            y = f(x)

where x and y are real numbers.

In vector analysis we compute integrals of vector functions of a real variable; that is we compute integrals of functions of the type

            F(t) = f₁(t) i + f₂(t) j + f₃(t) k

or equivalently,

where f₁(t), f₂(t), and f₃(t) are real functions of the real variable t.

Indefinite integral. Let F(t) = f₁(t) i + f₂(t) j + f₃(t) k be a vector depending on a single real variable t, where f₁(t), f₂(t), and f₃(t) are assumed to be continuous in a specified interval. By definition, the indefinite integral of F(t), if it exists, is given as

1)        

or equivalently,

where c₁, c₂ and c₃ are constants. It exists if and only if there exists some vector G(t) such that

It will exist if the terms on the right side of 1) are integrable.

The indefinite integral can also be defined as a limit of a sum in a manner analogous to that of elementary integral calculus.

Definite integral. If the integral of the function F(t) exists, let it be denoted by G(t) + c. Then the definite integral of F(t) between the limits t = a and t = b is given by

2)          

Example. The acceleration of a particle at any time t 0 is given by

            a = dv/dt = 12 cos 2t i - 8 sin 2t j + 16 t k .

Find the velocity at time t, if v = 0 at t = 0.

Integrating

                        = 6 sin 2t i + 4 cos 2t j + 8t² k + c₁

Putting v = 0 when t = 0,

            0 = 0i + 4j + 0k + c₁ .

Thus

            c₁ = 4j

and

            v = 6 sin 2t i + (4 cos 2t - 4) j + 8t² k

Line integrals.

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There are different types of line integrals. In general, any integral that is evaluated along a curve is called a line integral. Such integrals can be defined in terms of limits of sums as are the integrals of elementary calculus.

Types of line integrals.

where is a scalar point function of x, y, z and R is a radius vector R(t) = x(t) i + y(t) j + z(t) k defining curve C.

where F(x, y, z) is a vector point function of x, y, z, R is a radius vector R(t) = x(t) i + y(t) j + z(t) k defining curve C and the product is the dot product.

where F(x, y, z) is a vector point function of x, y, z, R is the radius vector R(t) = x(t) i + y(t) j + z(t) k defining curve C and the product is the cross product.

We now define line integrals of Type 1 above in terms of a limit of sums.

Def. Line integral of type

where f(x, y, z) is a scalar point function of x, y, z and R is a radius vector R(t) = x(t) i + y(t) j + z(t) k defining curve C.

Let C be a rectifiable space curve (i.e. a curve of finite length) in xyz-space, let AB be an arc on C and let f(x, y, z) be a function continuous at all points on the arc. Subdivide arc AB into n parts by means of points A = P₀, P₁, ... , P_n = B selected arbitrarily along it and let the points be connected by chords. Let Δ_is be the length of the chord joining P_{i -1} and P_i and let (x_i, y_i, z_i) be an arbitrarily chosen point on the segment of curve between P_{i -1} and P_i. See Fig. 1. Let Δ be the length of the longest chord. Form the sum

Let the number of subdivisions n approach infinity in such a way that the length of the longest chord approaches zero. The sum S_n will then approach a limit which does not depend on the mode of subdivision and is called the line integral of f(x, y, z) from A to B along the curve:

Line integrals are usually stated in the language of vectors. Curves are usually defined parametrically. Let us restate the above in terms of vectors with a parametrically defined curve. Let arc AB be defined by the radius vector R(t) = x(t) i + y(t) j + z(t)k on the interval from t = a to t = b and let f(x, y, z) be a scalar point function continuous at all points on the arc. Let {a = t₀, t₁, ..., t_n = b} be a partition of interval [a, b]. Let be a point of the interval [t_{i -1}, t_i] and let Δ be the largest of the intervals [t_{i -1}, t_i]. Form the sum

where

Let the number of subdivisions n approach infinity in such a way that the length of the longest chord approaches zero. The sum S_n will then approach a limit which does not depend on the mode of subdivision and is called the line integral of f(x, y, z) from A to B along the curve: 

Integral s_c Pdx + Qdy + Rdz .

Q. The integral ∫_C Pdx + Qdy + Rdz is a line integral. What does it mean? Where does it come from? How do we understand it?

A. Consider the line integral

where is a vector point function

defined over some region W of xyz-space, is a position vector

to a point P on a curve C within region W, the product is the dot product and the integral is evaluated along C. is given by

and

can be written as

The answer to the question is the integral ∫_C Pdx + Qdy + Rdz corresponds to ∫_C A₁(x, y, z)dx + A₂(x, y, z)dy + A₃(x, y, z)dz in the above and thus represents an integral of type

Some comments on line integrals. It is important to keep in mind that line integrals are different in a basic way from the ordinary integrals we are familiar with from elementary calculus. A line integral cannot be evaluated just as is. To evaluate it we need additional information — namely, the curve over which it is to be evaluated. This curve is generally given in parametric form such as

            x = x(t)

            y = y(t)

            z = z(t) .

The integral is then altered when substituting in the parametric equivalents and is transformed into an ordinary integral in the variable t and is evaluated as any ordinary integral. There are some forms of line integrals that look a lot like ordinary integrals. For example,

could be either line integrals or ordinary integrals and we would have to know which they were from context. Also, the integral

looks a lot like an ordinary integral and we need to realize that it is a line integral and that we need to have a parametrically defined curve over which to integrate it. And realize that its appearance will change and its meaning become clearer after substituting in the parametric equivalents.

Application in physics. Line integrals of type 2 above are of special interest in the field of physics. Let C be a space curve running from some point A to another point B in some region Q. Let C be defined by the radius vector R(s) = x(s) i + y(s) j + z(s) k where s is the distance along the curve measured from point A. Let F(x, y, z) = f₁(x, y, z) i + f₂(x, y, z) j + f₃(x, y, z) k represent a force field defined over the region. Let T denote the unit tangent to the curve at point (x, y, z). Then F∙T is the component of F in the direction of the tangent T and the integral

represents the work done in moving a body from point A to point B along curve C. The unit vector T is equal to

            T = dR/ds = dx/ds i + dy/ds j + dz/ds k .

so

            F∙T = (f₁ i + f₂ k + f₃ j)∙(dx/ds i + dy/ds j + dz/ds k )

                    =  f₁(dx/ds) + f₂(dy/ds) + f₃(dz/ds)

and 

Because T = dR/ds we can write

which is the form of 2] above. The quantity F∙Tds can be represented as g(x, y, z)ds where g(x, y, z) is a scalar point function that represents the component of F in the direction of the tangent of the curve. Thus

and from this we can note that type 2 above is really a special case of type 1.

From equation 3) above we state the following theorem:

Theorem 1. Let C be a space curve running from some point A to another point B in some region Q. Let curve C be defined by the radius vector R(s) = x(s) i + y(s) j + z(s) k where s is the distance along the curve measured from point A. Let F(x, y, z) = f₁(x, y, z) i + f₂(x, y, z) j + f₃(x, y, z) k represent a force field defined over the region. Let T denote the unit tangent to the curve at point (x, y, z). Then

where

represents the work done in moving an object along the curve from point A to point B.

● Since dR = dx i + dy j + dz k we can write

Suppose now that we have a space curve C defined by the parametric equations

            x = x(t)

            y = y(t)

            z = z(t),

or in vector form, R(t) = x(t) i + y(t) j + z(t) k , where t is time, and a force field defined by F(x, y, z) = f₁(x, y, z) i + f₂(x, y, z) j + f₃(x, y, z) k , in some region Q. We wish to compute the integral of

from point A to point B on curve C. If point A corresponds to time t = t₁ and point B corresponds to time t = t₂, then the integral becomes

where the primes denote derivatives with respect to t.

The line integral s_c f(x, y) dx.

Q. An integral of type ∫_C f(x, y) dx or ∫_C f(x, y) dy is regarded as a line integral and requires a curve for evaluation but it is different than other line integrals. What does it mean? Where does arise? How do we understand it?

A. One important place where this integral arises is in connection with evaluating the right member of the line integral

where is a vector point function

defined over some region R of the xy plane, is a position vector

to a point P on a curve C within region R, the product is the dot product and the integral is evaluated along C. is given by

The integral ∫_C f(x, y) dx is evaluated over an interval [a, b] of the x axis. The function f(x, y) is a scalar point function whose value varies with positions along the curve. To evaluate this integral it is tacitly assumed that the curve C is expressed as a single-valued function y = f(x) on the interval [a, b]. If the curve is like that shown in Fig. 3, it would have to be broken into two parts, one part expressed as y = f₁(x) [shown in blue in the figure] and the other expressed as y = f₂(x) [shown in red], and each part would have to be evaluated separately. What is the quantity that this integral computes? If we subdivide the interval [a, b] into n equal parts and let x_i correspond to the midpoint of the i-th subdivision, the integral is given by

where the general term is f(x_i, y_i)Δx_i. What intuitive meaning does this quantity have (this general term)? Why would one be interested in computing a sum of these quantities? The answer: It is an abstract quantity that arises in different contexts.

Surface integrals.

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Let S be a surface and let (P) be a point function defined on S. Divide S into a number of surface elements ΔS₁, ΔS₂, ... , ΔS_n of areas ΔA₁, ΔA₂, ... , ΔA_n. Let P_i be some point in the element ΔS_i. Then the surface integral of over S is defined as

where the limit is taken as the maximum of the dimensions of the elements ΔS_i approaches zero.

If (P) is expressed as a function F(x, y, z), where (x, y, z) are the coordinates of P, the surface integral becomes

Unit normal, n. Let S be a two-sided surface such as the one shown in Fig. 4. Let one side of S be considered arbitrarily as the positive side (if S is a closed surface this is taken as the outer side). A unit normal n to any point of the positive side of S is called a positive or outward drawn normal. 

Def. Vector dS = n dS. Let us associate with the differential of surface area dS a vector which we define as dS = n dS, whose magnitude is dS and whose direction is that of n.

Types of surface integrals. Let be a scalar point function and A be a vector point function. The following are types of surface integrals:

The integral of type 3 is of particular interest. It represents an integral of the flux A over a surface S.

To evaluate surface integrals we express them as double integrals taken over the projected area of the surface S on one of the coordinate planes. It is possible to do this if any line perpendicular to the coordinate plane chosen meets the surface in no more than one point. In general, this does not pose a problem since we can generally subdivide S into surfaces which do satisfy the restriction.

Volume integrals.

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Consider a closed surface in space enclosing a volume V. Volume integrals can be defined in terms of limits of sums in the same way as was done with surface integrals above.

Let S be a closed surface enclosing a volume V. Let (P) be a point function defined on V. Divide V into a number of volume elements ΔV₁, ΔV₂, ... , ΔV_n of volumes Δv₁, Δv₂, ... , Δv_n . Let P_i be some point in the element ΔV_i. Then the volume integral of over V is defined as

where the limit is taken as the maximum of the dimensions of the elements ΔVi approaches zero.

Types of volume integrals. Let be a scalar point function and A be a vector point function. The following are types of volume integrals:

References.

  Spiegel. Vector Analysis.

  Hsu. Vector Analysis.

  Wylie. Advanced Engineering Mathematics.

  Taylor. Advanced Calculus.