Multiple Integrals and Vector Calculus_Univ.leeds

8/12/2019 Multiple Integrals and Vector Calculus_Univ.leeds

1/42

MATH2420

Multiple Integrals and Vector Calculus

Prof. F.W. NijhoffSemester 1, 2007-8.

Course Notesand

General Information

Vector calculus is the normal language used in applied mathematics for solving problems in two and

three dimensions. In ordinary differential and integral calculus, you have already seen how derivativesand integrals interrelate. A derivative can be used as the opposite of an integration; it also occurs inchanging variables in an integral. The same interrelation applies in multiple dimensions, but with morerichness and variety.

This module starts with a discussion of different coordinate systems in two and three dimensions.The use of Cartesian, plane polar, cylindrical polar and spherical polar coordinates will run through thewhole module.

The second section starts with a discussion of vector functions, which are the two- and three-dimensional equivalents of the functions of ordinary calculus. These can be used to describe curvesin space. Next we look at functions of several variables: that is, functions of a vector. With these twoconcepts we can introduce derivatives for fully three-dimensional functions (gradient, divergence andcurl).

This brings us to the halfway point of the module, and we will pause to review our new understandingbefore moving on to multiple-dimensional integrals. Here we extend the familiar idea of integration inone dimension to integration over an area or a volume.

Finally, with the introduction of line and surface integrals we come to the famous integral theoremsof Gauss and Stokes. These encompass beautiful relations between line, surface and volume integralsand the vector derivatives studied at the start of this module.

Most real-life problems are not one-dimensional. The amount of heat stored in a piece of metal canbe calculated by integrating its temperature in three dimensions; and the diffusion of dye in water isgoverned by differential equations based on three-dimensional derivatives. This is why a knowledge ofvector calculus is essential for further study in many areas of applied mathematics.


2/42

1


3/42

Chapter 0

REVIEW

0.1 Calculus

Differentiation

The curvey = f(x) has a slope at pointx = a given by the derivativeoffwith respect to x at a:

f(a) = df

dx

a

= limx0

f(a+ x) f(a)

x . (1)

A few particular derivatives are:

f(x) =axn f(x) =anxn1

f(x) =ex f

(x) =ex

f(x) =u(x)v(x) f(x) =u(x)v(x) +u(x)v(x)

f(x) =u(x)/v(x) f(x) = [u(x)v(x) u(x)v(x)]/v2(x).

Integration

Integration is the opposite of differentiation: ba

f(x) dx= [f(x)]ba = f(b) f(a) or

f(x) dx= f(x) (2)

and it may also be seen as giving the area under a curve so the integral b

af(x) dxgives the area under

the curvey = f(x) between x = a andx = b. Some specific integrals are: xn dx= xn+1/(n + 1) forn =1

x1 dx= ln x

ex dx= ex/

u(x)v(x) dx= [u(x)v(x)]

u(x)v(x) dx

0.2 Lines and Circles

The vector equation of a straight line in three-dimensional space is

x= a +ub with u (, ) a real scalar. (3)

2


4/42


5/42


6/42

Course Notes 5

x

Figure 1.1: Spherical polar coordinates


7/42


8/42

Chapter 2

VECTOR CALCULUS

2.1 Vector Functions

Avector-valued functionf is a vector functionwhose components are single-valued functions(scalarvalued functions).

For example, given three single-valued functions f1(t), f2(t), f3(t) we can form the vector-valuedfunction

f(t) = (f1(t), f2(t), f3(t)) =f1(t)i+f2(t)j+f3(t)k (2.1)

Themagnitudeof the vector-valued function f(t) is a scalar-valued function and is defined by

||f(t)||=

f21 (t) +f

22 (t) +f

23 (t)

1/2

(2.2)

In general, the graph of the vector function f(t) =f1(t)i+ f2(t)j+f3(t)k is a curve C, in the sensethat, as t varies, the tip of the position vector f(t) traces out C. The equations

x= f1(t), y= f2(t), z = f3(t) (2.3)

corresponding to the components offare the parametric equations ofC. If one of the components iszero, e.g. f(t) =f1(t)i+f2(t)j, then C is said to be a planar curve, otherwiseC is a space curve.

2.1.1 Derivative of a vector function

Given the vector function f(t) = (f1(t), f2(t), f3(t)) the derivative offis defined by

f(t) = (f1(t), f

2(t), f

3(t)).

Properties of the derivative

(f+ g)(t) =f(t) +g(t)

(f)(t) =f(t) where is a constant scalar

(uf)(t) =u(t)f(t) +u(t)f(t), whereu is a scalar function

(f g)(t) =f(t) g(t) +f(t) g(t).

(f g)(t) =f(t) g(t) +f(t) g(t).

(f(u))(t) =f(u(t))u(t).

7


9/42

Course Notes 8

2.1.2 Integral of a vector function

Given the vector function f(t) = (f1(t), f2(t), f3(t)), the integral of f is defined bybaf(t) dt =

(ba

f1(t) dt,ba

f2(t) dt,ba

f3(t) dt).

Properties of the integral

ba

(f(t) +g(t)) dt=ba

f(t) dt+ba

g(t) dt

ba (f)(t) dt=

ba f(t) dt, where is a constant scalar

ba

(c f)(t) dt= c(t) ba

f(t) dt, wherec is a constant vector.

b

a(c f)(t) =c(t)

b

af(t) dt, where c is a constant vector.

2.2 Curves

1. The equation ofa straight line is parametrised by

r(t) =r0+td, t (, ) (2.4)

2. More generally, every vector function

r(t) =x(t)i+y(t)j+z(t)k (2.5)

parametrises acurve in space(or acurve in plane if one of the components x(t), y(t) orz(t) is

zero). It is also important to understand that a parametrised curveC is an orientedcurve in thesense that as t increases on some interval of definition I, the tip of the position vector r(t) tracesoutCin a certain direction. For example, the unit circle parametrised by

r(t) = cos(t)i+ sin(t)j, t [0, 2)

is traversed in the anticlockwise direction, starting at the point (1,0).

2.2.1 Tangent vector, tangent line

For a given curveCparametrised by

r(t) = (x(t), y(t), z(t)),

the derivative vectorr(t) = (x(t), y(t), z(t))

is called the tangent vectorto the curve Cat the point P(x(t), y(t), z(t)), and r(t) points out in thedirection of increasing t.

For a given curve Cparametrised byr(t) = (x(t), y(t), z(t)) the tangent lineat a pointt is the vectorfunction

R() =r(t) +r(t), (, ) (2.6)


10/42

Course Notes 9

2.2.2 Intersecting curves

Two curves

(C1) : r1(t) =x1(t)i+x2(t)j+x3(t)k,

(C2) : r2(u) =y1(u)i+y2(u)j+y3(u)k,

intersect iff there are numbers t and u for which r1(t) = r2(u). The angle between two intersectingcurves (which, by definition, is the angle between the corresponding tangent lines) can be obtained byexamining the tangent vectors at the point of intersection.

2.2.3 The unit tangent

For a given curveCparametrised by r(t) = (x(t), y(t), z(t)), the vector

T := r

(t)||r(t)||

(2.7)

is called the unit tangent vectorto the curve Cat the point P(x(t), y(t), z(t)). The unit tangent pointsin the direction of increasing t along the curve and is parallel to the curve.

2.2.4 Reversing the sense of a curve

We make a distinction between the curve

r= r(t), t [a, b] (2.8)

and the curveR(u) =r(a+b u), u [a, b]. (2.9)

Both vector functions trace out the same set of points, but the order has been reversed. Whereas thefirst curve starts at r(a) and ends at r(b), the second curve starts at r (b) and ends atr(a).

Thus, for example, the vector function

r(t) = cos(t)i+ sin(t)j, t [0, 2)

gives the unit circle traversed anticlockwise while the reversed curve

R(u) = cos(2 u)i+ sin(2 u)j, u [0, 2)

gives the unit circle traversed clockwise.

Unit tangent

When we reverse the sense of a curve, the unit tangent Treverses direction (is multiplied by1) becauseit always points in the direction of increasing t or u.

2.3 Arc Length

The length of a continuously differentiable curve (C): r= r(t), t [a, b] is given by

L(C) =

ba

||r(t)|| dt. (2.10)


11/42

Chapter 3

FUNCTIONS OF SEVERALVARIABLES

3.1 Introduction

Since we live in a three-dimensional world, in applied mathematics we are interested in functions whichcan vary with any of the three space variables x,y ,z and also with timet. For instance, if the functionfrepresents the temperature in this room, thenfdepends on the location (x,y,z) at which it is measuredand also on the time t when it is measured, so fis a function of the independent variables x, y, z andt, i.e. f(x , y , z , t).

3.2 Geometric InterpretationFor a function of two variables, f(x, y), consider (x, y) as defining a point P in the xy-plane. Let thevalue of f(x, y) be taken as the length P P drawn parallel to the z-axis (or the height of point P

above the plane). Then asPmoves in the xy-plane, P maps out a surfacein space whose equation isz = f(x, y).

Example: f(x, y) = 6 2x 3yThe surfacez = 6 2x 3y, i.e. 2x + 3y+z = 6, is a planewhich intersects thex-axis wherey = z = 0, i.e. x = 3;which intersects they-axis where x = z = 0, i.e. y = 2;which intersects thez-axis where x = y = 0, i.e. z = 6.

Example: f(x, y) =y2 x2In the planex = 0, there is a minimumat y = 0; in the plane y = 0, there is a maximumat x = 0.The whole surface is shaped like a horses saddle; and the picture shows a structure for which (0, 0)is called a saddle point.

3.2.1 Plane polar coordinates

Since the variables x and y respresent a point in the plane, we can express that point in plane polarcoordinates simply by substituting the definitions:

x= r cos y = r sin .

10


12/42


13/42


14/42

Course Notes 13

The order of mixed derivatives is not important.

Iff(x, y) represents a surface above the xy plane thenf/xis the slope of a section taken in thex direction at a pointP and f/y the slope of a section in the y direction. Between them thesetwo tangents define a plane which (we will see later) is the tangent planeto the surface at P.


15/42

Chapter 4

GRADIENTS

In this chapter we introduce derivatives in multiple dimensions. Although we limit ourselves to differ-entiating known functions, the real use of these derivatives is in differential equations. When we learntcalculus, there was a big gap between starting to study differentiation and solving our first differentialequation. In this chapter we meet three-dimensional derivatives. Because this is a major concept wewont get to any physical applications; but almost no physical problem can be solved without using thesetools.

4.1 Gradient and Directional Derivative

For a function of three variables f(x,y,z), the partial derivatives fx , fy and

fz measure the rates of

change of f along x, y and z directions, respectively. We now ask how we can calculate the rate of

change offin any direction in space. The answer lies in the vector

f=

f

x, f

y, f

z

=

f

xi+

f

yj+

f

zk (4.1)

call the gradientoff. From its definition, the component off alongi is (f i) = fx = rate of changealongi and similarly for the y and z directions. But foranydirection in space we are free to temporarilycall it the i-direction and carry over the above analysis. Thus in general:

For any direction in space defined by a unit vector u the rate of change off along u is given by(f u) and is called the directional derivativeoff alongu.

4.1.1 Two further properties of the gradient

We look at cases whereu is parallel or perpendicular to f. The changedf in fdue to a change in theposition P bydr = dru is given by change in f= (rate of change with distance) (distance), i.e.

df= (f u)dr= f (udr) =f dr= ||f||||dr|| cos() (4.2)

where is the angle between the vectors dr and f. From this equation it can be seen that the directiondr for which df is amaximumis obviously that for which cos() = 1, or = 0, i.e. the direction off.Thus

Property 1. At any point, fpoints in the direction in which fis increasing most rapidly and itsmagnitude||f|| gives this maximum rate of change.

Again from eq. (4.2), df= 0 corresponds to = /2, whenf anddr are perpendicular. Butdf= 0means that fhas not changed - so the displacement dr is along the surface f= const.. Thus

Property 2. At any point, fis perpendicular to the surfacef= const. through that point.

14


16/42

Course Notes 15

4.2 Linear Approximations (Tangents)

Motivation: Many functions arising in applications are difficult to deal with. We thus need ways ofapproximatingsuch functions by others which are easier to handle. The most useful approximations arepolynomials. We consider first the single variable case, f(x), which will guide us into the treatment ofthe two-variable case.

4.2.1 One-variable case (tangent line)

The tangent to the curve y = f(x) atA where x = a has the slopef(a) and therefore has the equationy = f(a)x+ const.. It passes throughA, i.e. forx= a, y = f(a), so const. = f(a) f(a)a. Thus theequation of the tangent line (the line parallelto the curve) is

y= f(a) + (x a)f(a). (4.3)

For points close to A the tangent gives a close approximation to the curve. The approximation

f(x) f(a) + (x a)f(a) (4.4)

is called the linear approximationto f(x) nearx = a.

4.2.2 Two-variables case (tangent plane)

Above we saw that, for a function of one variable, approximating its curve by a tangent line gave a linearapproximation. For a function of two variables, f(x, y), the corresponding linear approximation ariseswhen we approximate a surface by its tangent plane. Suppose the surface isz = f(x, y). The tangentplane at A is perpendicular to the normal at A, i.e. is perpendicular to the gradient vector at A. Now

the surface is f(x, y) z = 0, or F(x,y,z) = const. = 0, whose gradient is F = f

x i+ f

yj k. Thusthe tangent planer n= const. at A is

x

f

x

A

+y

f

y

A

z= const. (4.5)

This plane passes throughA(a, b) so the constant in eq. (4.5) has the value afx

A

+ bfy

A

f(a, b)

and thus (4.5) becomes

z = f(a, b) + (x a)

f

x

A

+ (y b)

f

y

A

. (4.6)

Thus the linear approximationto the surface close to A(a, b) is given by the tangent plane (4.6), so

f(x, y) f(a, b) + (x a)

f

x

A

+ (y b)

f

y

A

(4.7)

and we note that we can rewrite this using the gradient f as

f(x, y) =f(a, b) + ((x a), (y b)) (f)A. (4.8)


17/42


18/42


19/42


20/42


21/42


22/42

Course Notes 21

4.5.1 Jacobians of the standard coordinate transformations

The following Jacobians may be quoted as standard results:

(x, y)

(r, ) = r for plane polar coordinates

(x,y,z)

(r,,z) = r for cylindrical polar coordinates

(x,y,z)

(,,) = 2 sin for spherical polar coordinates.


23/42

Chapter 5

DOUBLE AND TRIPLEINTEGRALS

5.1 Multiple-Integral Notation

Previously ordinary integrals of the formJ

f(x) dx=

ba

f(x) dx (5.1)

whereJ= [a, b] is an interval on the real line, have been studied. Here we study double integrals

f(x, y) dx dy (5.2)

where is some region in the xy -plane, and a little later we will study triple integrals T

f(x,y,z) dx dy dz (5.3)

whereT is a solid (volume) in the xy z-space.

5.2 Double Integrals

5.2.1 Properties

(1) Area property

dx dy = Area of .

In particular if is the rectangle = [a, b] [c, d] then

dx dy = (b a)(d c).

(2) Linearity

[f(x, y) +g(x, y)] dx dy =

f(x, y) dx dy+

g(x, y) dx dy (5.4)

where andare constants.

22


24/42


25/42


26/42

Course Notes 25

5.3 Triple Integrals

5.3.1 Properties

(1) Volume property T

dx dy dz = Volume ofT .

In particular ifT is the box T = [a, b] [c, d] [e, f] then

T dx dy dz= (b a)(d c)(f e).

(2) Linearity

T

[f(x,y,z) +g (x,y,z)] dx dy dz

=

T f(x,y,z) dx dy dz+

T g(x,y,z) dx dy dz (5.14)

where andare constants.

(3) Additivity IfTis broken up into a finite number of nonoverlapping basic regions T1, . . . ,Tn, then T

f(x,y,z) dx dy dz =

T1

f(x,y,z) dx dy dz + . . .+

Tn

f(x,y,z) dx dy dz.

(5.15)

5.3.2 The Evaluation of Triple Integrals by Repeated Integrals

Let T be a solid whose projection onto the xy-plane is labelled xy. Then the solidT is the set of all

points (x,y,z) satisfying(x, y) xy, 1(x, y) z 2(x, y). (5.16)

The triple integral over Tcan be evaluated by setting

T

f(x,y,z) dx dy dz =

xy

2(x,y)1(x,y)

f(x,y,z) dz

dx dy. (5.17)

In eq. (5.17) we can evaluate the integration with respect toz first and then evaluate the double integralover the domain xy as studied for double integrals. In particular if xy is horizontally simple, say

a x b, 1(x) y 2(x). (5.18)

then the solid T itself is the set of all points (x,y,z) such that

a x b, 1(x) y 2(x), 1(x, y) z 2(x, y) (5.19)

and the triple integral over Tcan be expressed by three ordinary integrals as:

T

f(x,y,z) dx dy dz =

ba

2(x)1(x)

2(x,y)1(x,y)

f(x,y,z) dz

dy

dx. (5.20)

Here we first integrate with z [fromz = 1(x, y) toz = 2(x, y)], then with respect to y [fromy = 1(x)to y = 2(x)], and finally with respect to x [from x = a to x= b].


27/42

Course Notes 26

There is nothing special about this order of integration. Other orders of integration are possible and

in some cases more convenient. Suppose for example that the projection of T onto the xz-plane is adomain xz of the formz1 z z2, 1(z) x 2(z). (5.21)

IfTis the set of all (x,y,z) with

z1 z z2, 1(z) x 2(z), 1(x, z) y 2(x, z) (5.22)

then T

f(x,y,z) dx dy dz =

z2z1

2(z)1(z)

2(x,z)1(x,z)

f(x,y,z) dy

dx

dz. (5.23)

In this case we integrate first with respect to y, then with respect to x, and finally with respect to z.Still four other orders of integration are possible.

5.3.3 Evaluating Triple Integrals Using Cylindrical Coordinates

Let T be a solid whose projection onto the xy-plane is labelled xy. Then the solidT is the set of allpoints (x,y,z) satisfying

(x, y) xy, 1(x, y) z 2(x, y). (5.24)

The domain xy has polar coordinates in some set r and then the solid Tin cylindrical coordinatesis some solid S satisfying

(r, ) r , 1(r cos(), r sin()) z 2(r cos(), r sin()). (5.25)

Then

T

f(x,y,z) dx dy dz =

xy

2(x,y)1(x,y)

f(x,y,z) dz

dx dy

=

r

2(r cos(),r sin())1(r cos(),r sin())

f(r cos(), r sin(), z) dz

r dr d=

S

f(r cos(), r sin(), z)r dr d dz. (5.26)

5.3.4 Evaluating Triple Integrals Using Spherical Coordinates

Let Tbe a solid in xy z-space with spherical coordinates in the solid Sof-space. Then T

f(x,y,z) dx dy dz =

S

f( sin cos , sin sin , cos ) 2 sin d d d. (5.27)

5.4 Jacobians and changing variables in multiple integration

During the course of the last few sections you have met several formulae for changing variables in multipleintegration: to polar coordinates, to cylindrical coordinates, to spherical coordinates. The purpose ofthis section is to bring some unity to that material and provide a general description for other changesof variable.


28/42

Course Notes 27

5.4.1 Change of variables for double integrals

Consider the change of variables x = x(u, v) and y = y(u, v), which maps the points (u, v) of somedomain into the points (x, y) of some other domain . Then

The area of =

(x, y)(u, v) du dv. (5.28)

Suppose now that we want to integrate some function f(x, y) over . If this proves difficult to do directly,then we can change variables (x, y) to (u, v) and try to integrate over instead. Then

f(x, y) dx dy =

f(x(u, v), y(u, v))

(x, y)(u, v) du dv. (5.29)

5.4.2 Change of variables for triple integralsConsider the change of variables x = x(u,v,w), y = y(u,v,w), z = z(u,v,w) which maps the points(u,v,w) of some solid Sinto the points (x,y,z) of some other solid T. Then

The volume ofT =

S

(x,y,z)(u,v,w) du dv dw. (5.30)

Suppose now that we want to integrate some function f(x,y,z) over T. If this proves difficult to dodirectly, then we can change variables (x,y,z) to (u,v,w) and try to integrate over S instead. Then

T

f(x,y,z) dx dy dz =

S

f(x(u,v,w), y(u,v,w), z(u,v,w))

(x,y,z)

(u,v,w)

du dv dw. (5.31)

Referring back to equations (5.26) and (5.27), and the Jacobians given at the end of 4.5, we canverify that this formula is correct for a change from Cartesian to cylindrical coordinates (Jacobian isr)and for a change from Cartesian to spherical coordinates (Jacobian is 2 sin ).


29/42

Chapter 6

LINE INTEGRALS ANDSURFACE INTEGRALS

In this chapter we will study integration along curves and integration along surfaces. At the heart of thissubject lie three great theorems: Greens theorem, Gausss theorem(commonly known as the divergencetheorem) and Stokess theorem. All of these are ultimately based on the fundamental theorem of integralcalculus, and all can be cast in the same general form: An integral over a region S = An integral overthe boundary of S.

6.1 Line integrals

Leth(x,y,z) = (h1(x,y,z), h2(x,y,z), h3(x,y,z)) be a vector function that is continuous over a smoothcurveCparametrised by C :r(u) = (x(u), y(u), z(u)) withu [a, b]. The line integralofh overCis thenumber

C

h(r) dr=

ba

[h(r(u)) r(u)] du. (6.1)

Although we stated this definition in terms of three-dimensional vectorial functions h(x,y,z) and curvesin space r(u) = (x(u), y(u), z(u)), it also includes the two-dimensional case: h(x, y) and plane curvesr(u) = (x(u), y(u)).

If the curveCis not smooth but is made up of a finite number of adjoining smooth piecesC1, . . . , C n,i.e. it is piecewise smooth, then we define the integral over Cas the sum of the integrals over Ci fori= 1, . . . , n, that is

C

=C1

+ +Cn

. All polygonal paths are piecewise smooth.When we integrate over a parametrised curve, we integrate in the direction determined by the

parametrisation. If we integrate in the opposite direction, our answer is altered by a factor of 1,that isC

=C

.

6.1.1 Another notation for line integrals

Ifh(x,y,z) = (h1(x,y,z), h2(x,y,z), h3(x,y,z)) then the line integral over a curve Ccan be written asC

h(r) dr=

C

{h1(x,y,z) dx+h2(x,y,z) dy+h3(x,y,z) dz} . (6.2)

28


30/42

Course Notes 29

6.2 The Fundamental Theorem for Line Integrals

In general, if we integrate a vector function h from one point to another, the value of the line integraldepends on the path chosen. There is, however, an important exception. If the vector function h isa gradient, i.e. there exists a scalar function f such that h = f, then the value of the line integraldepends only on the endpoints of the path and not on the path itself. The details are spelled out in thefollowing theorem.

TheoremLet C, parametrised by r = r(u) with u [a, b], be a piecewise smooth curve that begins at= r(a) and ends at = r(b). Then if the vector function h is a gradient, i.e. h = f, we have

C

h(r) dr=

C

f(r) dr= f() f(). (6.3)

NOTE: It is important to see that this result is an extension of the fundamental theorem of

integral calculus:ba

f(x) dx= f(b) f(a).

CorollaryIf the curveCis closed, i.e. = , then f() =f() and

C

f(r) dr= 0.

6.3 Line integrals with respect to arc length

Suppose thatf is a scalar function continuous on a piecewise smooth curveCparametrised byr = r(u)withu [a, b]. Ifs(u) is the length of the curve from the tip ofr(a) to the tip ofr(u), then, as we haveseen in section 2.3, s(u) =||r(u)||. The integral off over C with respect to arc lengths is defined bysetting

C

f(r) ds=

ba

f(r(u))s(u) du. (6.4)

6.4 Greens Theorem

IfP(x, y) andQ(x, y) are scalar functions defined over a domain with piecewise smooth closed boundaryC, then

Q

x(x, y)

P

y(x, y)

dx dy =

C

P(x, y) dx+Q(x, y) dy (6.5)

where the integral on the right is a line integral over C taken in the anticlockwise direction.

Remark As indicated, the symbol

is used to denote the line integral over a simple closed curve Ctaken in the anticlockwise direction.

6.5 Parametrised Surfaces; Surface Area

We have seen that a space curve Ccan be parametrised by a vector function r = r(u) where u rangesover some interval Iof the u-axis. In an analogous manner, we can parametrise a surface Sin space bya vector functionr= r(u, v) where (u, v) ranges over some domain of the uv-plane.


31/42

Course Notes 30

Example (The graph of a function)

The graph of a function y = f(x), x [a, b] can be parametrised by setting r(u) = (u, f(u)),u [a, b].

Similarly, the graph of a function z = f(x, y), (x, y) can beparametrised by setting r(u, v) = (u,v,f(u, v)), (u, v) .

Example (A plane)If two vectors a and bare not parallel, then the set of all combinations ua +vb generates a planeP0 that passes through the origin. We can parametrise this plane by setting r(u, v) =ua +vb, u,v real numbers.

The planePthat is parallel to P0 and passes through the tip of a vector c can be parametrised bysetting r(u, v) =ua +vb +c, u,v real numbers.

Example (A sphere)

The sphere of radius a centred at the origin can be parametrised by setting

r(u, v) = (a sin(u)cos(v), a sin(u)sin(v), a cos(u)), (u, v) [0, ] [0, 2). (6.6)

6.5.1 The fundamental vector product

Let Sbe a surface parametrised by r = r(u, v), (u, v) . The cross product

N=ru r

v (6.7)

is called the fundamental vector productof the surface S.The vectorN(u, v) is perpendicular to the surface Sat the point with position vector r(u, v) and, if

different from zero, can be taken as the normal to the surface Sat that point.

ExampleFor the plane r(u, v) =ua +vb+c, the vector a bis normal to the plane.

ExampleThe fundamental vector product for a sphere is parallel to the radius vector r(u, v). (Using theparametrisation given above,N=a sin(u)r.)

6.5.2 The area of a parametrised surface

The area of a surface Sparametrised byr = r(u, v), (u, v) , is given by

Area ofS=

S

||N(u, v)|| du dv. (6.8)

Example (The surface area of a sphere)Using the parametrisation given by equation (6.6), we had N =a sin(u)r so ||N||= a2 sin(u) andthe area is

S

||N(u, v)|| du dv= a2 2v=0

u=0

sin(u) du dv= 2a2[ cos(v)]0 = 4a2.

Example (The area of a plane domain)A plane domain may be parametrised as r = (u,v, 0) for (u, v) . Then ru = (1, 0, 0) andrv = (0, 1, 0) and so the fundamental vector product is N = (0, 0, 1) which has magnitude 1.

1 du dv= Area of .


32/42

Course Notes 31

6.5.3 The area of a surface z=f(x, y)

Let the surfaceSbe the graph of the function z = f(x, y) with (x, y) . Then

Area ofS=

f

x

2+

f

y

2+ 1

1/2dx dy. (6.9)

In this case the parametrisation of S is r(u, v) = (u,v,f(u, v)), (u, v) and so N = (fx, fy, 1).The unit vectorn = N /||N|| is called the upper unit normal.

6.6 Surface Integrals

Let H(x,y,z) be a scalar function, continuous over a surface Sparametrised by r = r(u, v), (u, v) .The surface integralofH overSis the number

S

H(x,y,z) d=

H(r(u, v))||N(u, v)|| du dv. (6.10)

TakingH1 and referring back to eq. (6.8) we get S

d = Area ofS. (6.11)

6.6.1 Flux of a vector function

Letq(x,y,z) be a vector function that is continuous over a smooth surface Sparametrised byr = r(u, v),(u, v) . The fluxofqacross Sin the direction of the unit normal n to the surface Sis the number

S

q n d (6.12)

which can be calculated as S

q n d=

q(r(u, v)) n ||N|| du dv=

q(r(u, v)) Ndu dv. (6.13)

Proposition

IfSis the graph of a function z = f(x, y) with (x, y) and n is the upper unit normal, then the fluxof the vector function q= (q1(x,y,z), q2(x,y,z), q3(x,y,z)) across Sin the direction ofn is

S

q n d=

(q1fx q2fy+q3) dx dy. (6.14)

ProofWe can parametrise the surface by r = (u,v,f(u, v)) with (u, v) . Then the fundamental vectorproduct is

N=r u r

v = (1, 0, fu) (0, 1, fv) = (fu, fv, 1)

and we have S

q n d =

(q N) du dv

=

(q1fu q2fv+q3) du dv=

(q1fx q2fy+q3) dx dy.

where we have simply changed the names of the variables at the end.


33/42

Course Notes 32

6.7 The Divergence (Gauss) Theorem

Recall that ifP(x, y) and Q(x, y) are scalar functions defined over a domain with piecewise smoothclosed boundaryC, then Greens theorem (section 6.4) allowed us to express a double integral over asa line integral over C:

Q

x(x, y)

P

y(x, y)

dx dy=

C

P(x, y) dx+Q(x, y) dy. (6.15)

This formula can be rewritten in vector terms (usingq= (Q, P)) to give the divergence theorem in twodimensionsas follows:

The divergence theorem in two dimensionsLet be a two-dimensional domain bounded by a piecewise smooth closed curve C. Then for any

(continuously differentiable) vector function q(x, y) we have that

( q) dx dy=

C

(q n) ds (6.16)

wheren is the outer unit normal and the integral on the right is taken with respect to arc length.

We can now give the three-dimensional analogue of the divergence (Gauss) theorem.

The divergence theorem in three dimensionsLetT be a three-dimensional solid bounded by a piecewise smooth closed surfaceS. Then for any(continuously differentiable) vector function q(x,y,z) we have that

T( q) dx dy dz =

S(q n) d (6.17)

wheren is the outer unit normal.

6.7.1 Divergence as outward flux per unit volume

In eq. (6.17), the right-hand side

S(q n) d represents the q across S in the direction ofn. In this

sense, from eq. (6.17) we can say that the divergence is the outward flux per unit volume, as we discussedin section 4.4.1.

Points (x,y,z) T for which

q(x,y,z)< 0 are called sinks.

q(x,y,z)> 0 are called sources. If q(x,y,z) 0 then qis called solenoidal.

6.8 Stokess Theorem

We return to Greens theorem (section 6.4):

Q

x(x, y)

P

y(x, y)

dx dy=

C

P(x, y) dx+Q(x, y) dy. (6.18)


34/42

Course Notes 33

and this time setting q= (P,Q,R) a vector function, we have

( q) k= det

i j k

xy

z

P Q R

k= Q

x

P

y. (6.19)

Thus in vector terms Greens theorem can be written as

[( q) k] dx dy =

C

q(r) dr. (6.20)

Since any plane can be coordinatised as thexy-plane, this result can be phrased in the following theorem

Stokess theoremLet S be a smooth surface with smooth bounding curve C. Then for any (continuously differen-

tiable) vectorial function q(x,y,z) we have S

[( q) n] d =

C

q(r) dr (6.21)

where n is a unit normal that varies continuously on S, and the line integralC

is taken in thepositive sense with respect to n.

6.8.1 The normal component ofqas circulation per unit area; Irrotationalflow

Interpret the vector function q(x,y,z) as the velocity of a fluid. In eq. (6.21), the right-hand side line

integralCq(r) dr is called the circulationofqaround the curveC. In this sense, from eq. (6.21), wecan say that qin the direction n is the circulation ofqper unit area, which relates to the rotation

of the fluid as discussed in section 4.4.2.If q0 then there is no circulation and qis called irrotational, i.e. the fluid has no rotational

tendency.


35/42


36/42

Chapter 8

GENERAL INFORMATION

Lecturer: Prof. F. W. NijhoffDepartment of Applied MathematicsRoom 9.20a, telephone ext. 35120e-mail: [email protected]

Lectures and example classes There will be 33 hours total of lectures and classes during 11 weeks.

Lectures on Tuesdays 3-4pm in RSLT 8;

Classes on Mondays 4-5pm in RSLT 6;

Lectures on Fridays 1-2pm in RSLT 6.

webpage The MATH2420 webpage conatining most of the material can be found under the URL:http://www.maths.leeds.ac.uk/%7Efrank/math2420.html

or follow the link under the Lecturers staff page:http://www.maths.leeds.ac.uk/ frank

and click on Teaching. This page will contain most of the material, but be aware that some ma-terial may be updated during the term.

Example sheets Every two weeks you are expected to hand in the solutions to an example sheet. Yourwork will be marked to monitor progress on the first 5 sheets, but worked solutions to the lastsheet (on the last section of the course) will be handed out to help your revision over the Christmasvacation. The completed sheets will be handed in at the Friday lectures, on the dates given onpage 34. The schedule will be as follows:

Expl. sheet Handout date Due date# 1 25/9 5/10# 2 5/10 19/10# 3 19/10 2/11# 4 2/11 16/11# 5 16/11 30/11# 6 27/11 not marked

Marking The exercises will be marked by postgraduate students (the top mark being 5 for each sheet).They count for up to 15% of your course marks.

35


37/42

General Information 36

What I expect from you

Attend lectures and examples classes

Take notes during lectures (this document is only a summary)

Attempt the examples on your own

Hand in example sheets on time

Ask questions as they occur to you

Turn off your mobile phone!

be considerate to your fellow students (do not hold disruptive conversations during lectures)

Booklist

1. M. R. Spiegel, Theory and Problems of Vector Analysis, McGraw-Hill.

2. E. Kreyszig, Advanced Engineering Mathematics, Wiley.

3. P. V. ONeil, Advanced Engineering Mathematics, PWS-Kent Publishing.

4. A. C. Bajpai, Advanced Engineering Mathematics, Wiley, 1977.

5. C. R. Wylie and L. C. Barrett, Advanced Engineering Mathematics, McGraw-Hill.

6. P. C. Matthews, Vector Calculus, 1998.


38/42


39/42


40/42

Contents

0 REVIEW 20.1 Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

0.2 Lines and Circles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.3 Trigonometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30.4 Determinant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30.5 Vector Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1 COORDINATE SYSTEMS 41.1 Cartesian Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2 Plane Polar Co ordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3 Cylindrical Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.4 Spherical Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 VECTOR CALCULUS 72.1 Vector Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.1.1 Derivative of a vector function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.1.2 Integral of a vector function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2 Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.2.1 Tangent vector, tangent line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.2.2 Intersecting curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2.3 The unit tangent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2.4 Reversing the sense of a curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.3 Arc Length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3 FUNCTIONS OF SEVERAL VARIABLES 103.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.2 Geometric Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3.2.1 Plane polar coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.3 Partial Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.3.1 Second-order partial derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

4 GRADIENTS 144.1 Gradient and Directional Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

4.1.1 Two further properties of the gradient . . . . . . . . . . . . . . . . . . . . . . . . . 144.2 Linear Approximations (Tangents) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

4.2.1 One-variable case (tangent line) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154.2.2 Two-variables case (tangent plane) . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

39


41/42


4.3 The Chain Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

4.3.1 Extended chain rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164.4 The vector differential operator (grad) . . . . . . . . . . . . . . . . . . . . . . . . . . . 164.4.1 Divergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174.4.2 Curl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184.4.3 Grad and div in polar coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . 184.4.4 Basic Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.4.5 The Laplacian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

4.5 Jacobian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204.5.1 Jacobians of the standard coordinate transformations . . . . . . . . . . . . . . . . 21

5 DOUBLE AND TRIPLE INTEGRALS 225.1 Multiple-Integral Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225.2 Double Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

5.2.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225.2.2 Geometric Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235.2.3 The Evaluation of Double Integrals by Repeated Integrals . . . . . . . . . . . . . . 235.2.4 Evaluating Double Integrals Using

Polar Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245.3 Triple Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

5.3.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255.3.2 The Evaluation of Triple Integrals by Repeated Integrals . . . . . . . . . . . . . . . 255.3.3 Evaluating Triple Integrals Using Cylindrical Coordinates . . . . . . . . . . . . . . 265.3.4 Evaluating Triple Integrals Using Spherical Coordinates . . . . . . . . . . . . . . . 26

5.4 Jacobians and changing variables in multiple integration . . . . . . . . . . . . . . . . . . . 265.4.1 Change of variables for double integrals . . . . . . . . . . . . . . . . . . . . . . . . 27

5.4.2 Change of variables for triple integrals . . . . . . . . . . . . . . . . . . . . . . . . . 27

6 LINE INTEGRALS AND SURFACE INTEGRALS 286.1 Line integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

6.1.1 Another notation for line integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . 286.2 The Fundamental Theorem for Line Integrals . . . . . . . . . . . . . . . . . . . . . . . . . 296.3 Line integrals with respect to arc length . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296.4 Greens Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296.5 Parametrised Surfaces; Surface Area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

6.5.1 The fundamental vector product . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306.5.2 The area of a parametrised surface . . . . . . . . . . . . . . . . . . . . . . . . . . . 306.5.3 The area of a surfacez = f(x, y) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

6.6 Surface Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

6.6.1 Flux of a vector function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316.7 The Divergence (Gauss) Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

6.7.1 Divergence as outward flux per unit volume . . . . . . . . . . . . . . . . . . . . . . 326.8 Stokess Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

6.8.1 The normal component of qas circulation per unit area; Irrotational flow . . . 33

7 EXAMPLE SHEETS 34

8 GENERAL INFORMATION 35

A GREEK ALPHABET 37


42/42


B IN POLAR COORDINATES 38

Documents

Multiple Integrals and Vector Calculus_Univ.leeds