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MATH2420 Multiple Integrals and Vector Calculus Prof. F.W. Nijhoff Semester 1, 2007-8. Course Notes and 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 derivatives and integrals interrelate. A derivative can be used as the opposite of an integration; it also occurs in changing variables in an integral. The same interrelation applies in multiple dimensions, but with more richness 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 the whole 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 curves in space. Next we look at functions of several variables: that is, functions of a vector. With these two concepts we can introduce derivatives for fully three-dimensional functions (gradient, divergence and curl). This brings us to the halfway point of the module, and we will pause to review our new understanding before moving on to multiple-dimensional integrals. Here we extend the familiar idea of integration in one dimension to integration over an area or a volume. Finally, with the introduction of line and surface integrals we come to the famous integral theorems of Gauss and Stokes. These encompass beautiful relations between line, surface and volume integrals and 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 can be calculated by integrating its temperature in three dimensions; and the diffusion of dye in water is governed by differential equations based on three-dimensional derivatives. This is why a knowledge of vector calculus is essential for further study in many areas of applied mathematics. 1 Chapter 0 REVIEW 0.1 Calculus Differentiation The curve y = f(x) has a slope at point x = a given by the derivative of f with respect to x at a: ′ df f(a+∆x)−f(a) f (a) = = lim : (1) dx ∆x→0 ∆x a Afew particular derivatives are: n ′ n−1 f(x) = ax f (x) = anx f(x) = eλx f′(x) = λeλx 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: Z b ′ b Z ′ f (x)dx = [f(x)] = f(b)−f(a) or f (x)dx = f(x) (2) a a and it may also be seen as giving the area under a curve – so the integral Rb f(x)dx gives the area under a the curve y = f(x) between x = a and x = b. Some specific integrals are: Z n n+1 Z −1 x dx=x =(n+1) for n 6= −1 x dx=lnx Z eλxdx = eλx=λ Z u(x)v′(x)dx = [u(x)v(x)] −Z u′(x)v(x)dx 0.2 Lines and Circles The vector equation of a straight line in three-dimensional space is x=a+ubwithu∈(−∞;∞)arealscalar: (3) 2 Course Notes 3 2 2 2 The equation of a circle of radius r centred about the origin is x + y = r and if the circle is centred 0 0 2 2 2 on the point P(a;b) the the equation is (x − a) +(y −b) = r . 0 0.3 Trigonometry Recall the functions sinx and cosx, with the identities: 2 2 sin x+cos x = 1 (4) tanx = sinx=cosx (5) and the derivatives (d=dx)sinx = cosx (6) (d=dx)cosx = −sinx (7) 2 (d=dx)tanx = 1+tan x: (8) There are memorable values: x sinx cosx 0 0 1 π=2 1 0 π 0 −1 3π=2 −1 0 2π 0 1 0.4 Determinant The determinant of a 3x3 matrix A is a11 a12 a13 |A| = a21 a22 a23 = a11(a22a33 −a32a23)−a12(a21a33 −a31a23)+a13(a21a32 −a31a22) (9) a31 a32 a33 0.5 Vector Products Weconsider two vectors a = (a1;a2;a3) and b = (b1;b2;b3). Their scalar dot product is given by a·b = a1b1 +a2b2 +a3b3 (10) and their vector cross product by i j k a×b= a1 a2 a3 =([a2b3−b2a3];[a3b1−b3a1];[a1b2 −a2b1]): (11) b1 b2 b3 Note that although for the scalar product a· b = b · a, for the vector product we have a × b = −b×a. Recall that a·b = ||a||||b||cosθ and ||a×b|| = ||a||||b||sinθ (12) where θ is the angle between the vectors a and b.
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