H EXAMPLES Mathematical Methods : Courses A and B: Michaelmas Term 4 Natural Sciences A, Computer Science A This sheet provides exercises for the whole of the Michaelmas Term. Questions marked * are suitable for Course A. Supervisors will advise about selection of questions. Answers will be circulated as term progresses. RHB (x.x/y.y) where applicable, gives reference to a problem in Riley, Hobson and Bence st edition (x.x) or nd edition (y.y). Comments and suggestions are welcome and can be sent by e-mail to rea@phy.cam.ac.uk. I.R. Parry, R.E. Ansorge : VECTORS A. Vector Addition and Subtraction *A An aeroplane has an air speed of 5 km/hr due North. How fast will it travel over the Earth, and in what direction, if the wind is: (a) From the North, speed 4 km/hr, (b) From the West, speed 5 km/hr, (c) From the South-east, speed 8 km/hr? *A An aeroplane has an air speed of 5 km/hr. The pilot wishes to travel km due North. What course should be set and how long will the journey take if the wind is: (a) From the North, speed 4 km/hr, (b) From the West, speed 5 km/hr, (c) From the South-east, speed 8 km/hr? *A3 Prove that a quadrilateral is a parallelogram if its diagonals bisect each other. *A4 (RHB 6./7.) The unit cell of diamond is a cube of side a with carbon atoms at each corner and at the centre of each face (i.e. face centred cubic ). In addition there are carbon atoms displaced by 4a(i + j + k) from each of the previously mentioned atoms, where i, j, k are unit vectors along the cube axes. One corner of the cube is taken as the origin of coordinates. What are the vectors joining the atom at 4a(i + j + k) to its four nearest neighbours? *A5 (RHB fig 6.6/7.6) ABC is a triangle whose vertices have position vectors a, b, c. (a) What is the position vector d of the mid-point of BC? (b) Write down the position vector p of a point P on the median joining A to the mid-point of BC and a fraction, λ, of the way along it. (c) Write down similar expressions for points on the other two medians. (d) By guessing a suitable value of λ (or otherwise) show that the three medians all meet in one point. What is its position vector? A6 Show that, for any tetrahedron, the lines joining the mid-points of opposite edges are concurrent. A7 (RHB 6.8/7.6) a, b and c are coplanar vectors related by λa + µb + νc = (λ, µ, ν are not zero). Show that the condition for the points with position vectors αa, βb, γc to be collinear is λ α + µ β + ν γ =.
*A8 (RHB 6.7/7.) Show that the points with position vectors,, 3 4 lie on a straight line and give the equation of the line in the two forms (a) r = a + λb (b) x x c = y y d = z z. e B. Scalar Product *B The points A and B have position vectors a = (, 3, 4) and b = (3,, ) relative to origin O. Find (a) The lengths of OA, OB and AB, (b) The angle AOB, (c) The angle between OA and AB. *B Four points are such that AH BC and BH AC. Deduce that CH AB. *B3 What are the properties of the plane b.(r a) =? Find (a) the length of the projection of OC onto the plane, and (b) the distance of C from the plane, where O is the origin and C has a position vector c. *B4 From the inequality (a.b) a b deduce the triangle inequality a + b a + b. *B5 (RHB 6./7.3) Identify the surfaces r = k, r.u = l, r.u = m r, r (r.u)u = n where k, l, m and n are fixed scalars and u is a fixed unit vector. *B6 Find a and b such that the vectors e =, e = a and e 3 = b form an orthogonal basis. With this choice of a and b find c, d and e if (c + d )e + (e )e + (c + e)e 3 =. *B7 (RHB 6.9a/7.7a) Show that the line of intersection of the two planes x + y + 3z = and 3x + y + z = is equally inclined to the x and z axes and makes an angle cos ( /3) with the y-axis.
*B8 Calculate the shortest distances between the plane 5x+y 7z +9 = and the points (,, 3) and (3,, 3). Are the points on the same or opposite sides of the plane? B9 Prove that for any tetrahedron the sum of the squares of the lengths of the edges equals four times the sum of squares of the lengths of the lines joining mid-points of opposite edges. B The vectors a, b, c are of equal length l, and define the positions of the points A, B, C relative to O. If OABC is a regular tetrahedron, find the distance of one vertex from the opposite face. *B Find the acute angle at which two diagonals of a cube intersect. C. Vector Product *C Find the cross product of the vectors i 3j + 5k and 3i + j k and show that it is perpendicular to both vectors. *C Find the angle between the position vectors of the points (3, 4, ) and (3,, 6), and find the direction cosines of a vector perpendicular to both. *C3 Show that for any three vectors a, b, c *C4 Show that for any three vectors a, b, c (b a) (c a) = a b + b c + c a. (a b) c + (b c) a + (c a) b = *C5 Describe the locus of points r that satisfy the equation where a = (,, ) and b = (,, ). r a = b, *C6 The vectors a, b, c are not coplanar. The vectors A, B, C, defined by A = b c [a, b, c] ; B = c a [a, b, c] ; C = a b where [a, b, c] = a b c [a, b, c] are said to be the reciprocal vectors. Show that (a) A.a = B.b = C.c = (b) A.b = A.c = B.a =, etc. (c) [A, B, C] = /[a, b, c] (d) a = B C/[A, B, C] C7 Show that, in the notation of the previous question, the plane through the points a/α, b/β, c/γ is normal to the direction of the vector g = αa + βb + γc. Show that the perpendicular distance of the plane from the origin is g. *C8 You need to drill a hole in a piece of metal, at right angles to a flat surface passing through the points A = (,, ), B = (,, ) and C = (,, ) and emerging at the point D = (,, ). How long a drill must you use, and where (in the plane ABC) must you start drilling?. 3
C9 Find the distance between any vertex of a unit cube and a diagonal of the cube which does not pass through it (a) using a cross-product (b) by finding the perpendicular from the vertex to the diagonal. *C Find an equation of the form (r a).n = for the plane which passes through the points (,, ), (,, 3) and (,, 4). *C Show that the vectors a =, b = and c = form a non-orthogonal basis and, using the triple scalar product, write the vector d = in terms of this basis. C Simplify (a + b).(b + c) (c + a) and (a b). ((b c) (c a)). C3 Solve the vector equation a r + λr = c for r, where λ. D. Vector Area *D A = (,, ), B = (,, ), C = (,, ), D = (,, ), E = (,, ), O = (,, ). What are the vector areas of: (a) the square OABC (b) the rectangle OAED (c) the upper surface of the pyramid with base OABC and vertex (,, ). (d) A lampshade (truncated hollow cone) bounded by a horizontal circle of radius 4 and a horizontal circle of radius 3, height 3 above the first. D O = (,, ), A = (,, ), B = (,, ), C = (,, ). Find s, the vector area of the loop OABCO (a) by drawing the loop in projection onto the yz, zx, and xy planes and thus finding the components of s, (b) by filling in the loop with two or three plane polygons, finding the vector area of each, and taking the resultant. What is the projected area of the loop seen (c) from the direction (,, ), (d) from the direction which makes the projected area as large as possible? E. Coordinate Systems *E A point has Cartesian coordinates (3, 4, 5). What are its coordinates in (a) spherical polars and (b) cylindrical polars? E Taking the z axis along the polar axis and the x axis in the meridian plane, calculate the Cartesian coordinates of a point in terms of (a) its spherical polar coordinates, (b) its cylindrical polar coordinates. 4
*E3 Identify the following loci, where a is constant. (a) Plane polars: () φ = a () r = φ (b) Cylindrical polars: () z = a () r = a (3) r = a and z = φ (c) Spherical polars: () θ = a () φ = a (3) r = a (4) r = θ = a (5) r = θ = φ. CALCULUS F. Revision of Calculus *F Find dx if y + ey sin y = x, (a) by differentiating the expression with respect to x, (b) by expressing x as a function of y and then finding dx. [Your answers for /dx should agree!] *F Sketch the following curves: (a) y = xe x, (b) y = x log x, (c) y = + x x +, (d) y = + e x, (e) y = e x, (f) y = ex cos x, (g) y = e cos x. *F3 Evaluate the integral partial fractions. 4 3x x + x dx by decomposing the integrand into *F4 With the help of suitable substitutions, find the indefinite integrals (a) tan θ sec θ dθ, e x dx (b), e x 8 x (c) dx. (6x x ) *F5 Using integration by parts, or otherwise, find the indefinite integrals (a) ln ( x 3) dx, (b) (ln x) 3 dx. x x *F6 Let I = e at cos bt dt, and J = e at sin bt dt, Using integration by parts, show that where a, b are constants. I = b e ax sin bx ab J. Find another similar relationship between I and J, and hence find I and J. 5
*F7 Show by means of a sketch that sin x π x, for x π. Hence show that π x dx ( + sin x < π3 π ) 8 4 *F8 State whether the following functions are even, odd or neither:. (a) x (b) (x a) (c) sin x (d) sin(π/ x) (e) exp x (f) x cos x (g) x (h) (i) ln[( + x)/( x)] F9 Tripos 999 () Explain what is meant by the statement that a function f(x)is (a) continuous at x = x, (b) differentiable at x = x. Sketch graphs of the following functions: (c) f(x) = x ; (d) f(x) = x ; { x sin (e) f(x) = x, x., x = For each of the functions, determine the range(s) of x for which f(x) is (f) continuous, (g) differentiable. F Approximate the following functions in the limit x. In each case give the leading term and the order of the error. x 3 + x x +, cos x x. F Approximate the following function in the limit x. Give the leading term and the order of the error. + x + x. 3x + 3 F Use the binomial expansion to approximate the following function in the limit x. ( x 4 x ) 4 ( x + x ). 6
F3 Use Leibniz s formula to establish that ( d n Z n = dx n is a solution of the equation ) e x / Z n + xz n + (n + )Z n =. G. Integration and Schwarz s Inequality G Tripos 998 () 6. Consider F (s) = +s f(x, s)dx. (a) Give an expression for df/ds. If f(x, s) = x(s x) then (b) Determine df/ds using the expression given for (a). (c) Calculate the integral F (s) and then differentiate with respect to s to confirm your answer to (b). (d) State the values of s for which df/ds =. G By considering the area under the curves y = log x and y = log(x ), show that Hence show that: N log N N < log(n!) < (N + ) log(n + ) N. ( log N! N log N + N < log + ) N + log( + N). N G3 A particle is located with probability p(x)dx between x and x + dx. If G4 If x n = x n p(x)dx, show, using Schwarz s inequality, that x x. I = π sin x x + dx, obtain an upper bound for I using (i) Schwarz s inequality, and (optional) (ii) the inequality x > sin x. Which method gives the better bound? 7
3. MULTIPLE AND GAUSSIAN INTEGRALS *H Evaluate the integral x= y=x/ xy dx. Change the order of integration (integrate with respect to x first) and thence evaluate the integral again. *H By evaluating the appropriate triple integrals find (a) the volume of a sphere of radius a; (b) the volume of a flat-topped cone, described, in cylindrical polar coordinates by {(r, φ, z) z a, φ π, r z}; (c) the volume of a round-topped cone, described, in spherical polar coordinates by {(r, θ, φ) r a, φ π, θ π 4 }. *H3 Integrate both sin(x + y) and xe xy (a) over the square < x < π/, < y < π/, (b) over the square < x < π, < y < π, (c) over the triangle formed by the lines y =, x = π, x = y. (Leave your answer in the form of a single integral in the one case in which it cannot be evaluated by standard means.) H4 Sketch the curve given by r = a( + cos φ) and evaluate where D is the interior of the curve. D (x + y ) / dx *H5 Evaluate the integral x ( x y )dx over the interior of the circle of radius centred at x =, y =. H6 Show that D x ydx = (3 )/4 where D is the smaller of the two areas bounded by the curves xy =, y = x and y =. *H7 By means of mathematical induction show that x n e ax dx = (n )(n 3)...3. (a) n { π a } / for integer n and positive a. *H8 Evaluate the volume integral of e (x +y +z )/a over the whole of three-dimensional space. 8
*H9 According to the Schrödinger Equation of Quantum Mechanics, an electron in the ground state of an hydrogen atom has a spherically symmetric spatial probability density distribution P (r) = Ke r/a, where r is the distance between the electron and the proton and the constant K is chosen to make the overall probability of finding the electron. Calculate (a) K. (b) The average value of r. (c) The most probable value of r. 4: POWER SERIES *I Find the first four terms in the Taylor expansion of sin x about x = π/6. Hence find an approximate value for sin 3. Justify the precision to which you answer. *I Find the first three non-zero terms of the Taylor series expansion about x = of *I3 Show that for x (a) sin x ; (b) tan x ; (c) / ( + x). tan x = x x 3 /3 + x 5 /5... (a) Using the result tan () = π/4, how many terms of the series are needed to calculate π to places of decimals? (b) Show that π/4 = tan (/) + tan (/3) and deduce another series for π. How many terms of this series are needed to calculate π to places of decimals? (c) Show that π/4 = 4 tan (/5) tan (/39) and deduce yet another series for π. How many terms of this last series are needed to calculate π to decimal places? I4 (RHB 3.8/4.5) Obtain approximate expressions for ln(a/b) when a and b are positive and nearly equal: ( ) + x (a) by using the logarithmic series for ln show that x ln(a/b) a b a + b and that the error in this approximation is (a b) 3 /a 3. (b) by using the logarithmic series for ln( + x) show that ln(a/b) (a b)[(/a) + (/b)] and that the error in this approximation is 4/3 ((a b)/(a + b)) 3. 9
*I5 Tripos 993 (). Newton-Raphson method. Write down the Taylor series for a function f evaluated at x + h in terms of f and its derivatives evaluated at x. Use the result to show that if x is an approximate solution of the equation f(x) =, then a better approximation is given in general by x = x, where x = x f(x ) f (x ). ( ) Sketch the graph of ln(x). Suppose ɛ is a fixed small positive number. By double application of ( ) or otherwise find the smaller solution of the equation ln(x) = ɛx correct to terms proportional to ɛ. 5: COMPLEX NUMBERS *J Find the real and imaginary parts of the following complex numbers, (n is an integer): (a) i 3 (b) i 4n (c) /i (d) (e) ( ) ( ( i) (f) + (i) 4 + i 3 + i (j) ( + i i ) 3 (g) + i 3i 3i ) (k) e iωt R + iωl + (iωc) *J Plot the following complex numbers on an Argand diagram: z = + i, z = 3 + 4i, z, z, z z, z z. ( ) + i (h) /( + i) Find z z (a) by adding arguments and multiplying moduli; and (b) by using the rules of complex algebra. *J3 Factorise the following expressions: (a) z +, (b) z z +, (c) z + i, (d) z + ( i)z i. J4 By taking the complex conjugate of the equation, show that the solutions to z + az + b = (a, b real) are either real or come in complex conjugate pairs. What can be said about n th order polynomial equations with real coefficients (i.e. equations of the form a + a z + a z +... + a n z n = where a, a... a n are real)? *J5 Find the modulus and principal argument of (a) + 3i ; (b) + i ; (c) 3 (i/ 3) ; (d) e iωt R+iωL+(iωC). *J6 If z = x + iy, find the real and imaginary parts of the following functions (a) z ; (b) iz ; (c) ( + i)z ; (d) z (z ).
*J7 Find the set of points in the complex plane described by the equations (a) z i = (b) (z i) = (c) (z i) = (d) z i = (e) z = z + i (f) z = z + i (g) z = z + i (h) z = z (i) z i = z (j) arg(z) = π (k) arg(z ) = π 4 (l) arg(z) = z z is the complex conjugate of z. J8 Describe the curve in the Argand diagram whose equation is z i = 6. As z moves round this curve, what are the loci of u = z + 5 8i, v = iz +? (Note that this problem does not require any calculations!) *J9 Show that z z is invariant under a rotation of z and z about the origin in the Argand diagram. J Show that a + b + a b = ( a + b ), where a and b are complex numbers. Interpret this result geometrically. J u and v are defined as the real and imaginary parts of the function w = /z where z = x + iy. Show that the contours of u = A = constant and v = B = constant in the complex z plane are circles with centres on the x and y axes respectively, whose radii and position depend on A and B. Find the radii and centres of the circles, and show that the circles of u = A cross those of v = B at right angles. *J (a) Find sin 3θ in terms of powers of sin θ. (b) Find sin 5 θ in terms of sin θ, sin θ, sin 3θ etc. J3 Calculate the following sums: (*a) 5 n= cos(nθ); (*b) N n= sin(nθ); (c) N n= rn cos(nθ). J4 Express log x in terms of log e x. J5 Find d dx (ax ) where a and x are real. *J6 Write the following in the form a + ib where a and b are real, (a) e iπ/ (b) e iπ (c) e iπ/4 (d) e +i (e) exp(e iπ/4 ) (f) exp(re iθ ).
*J7 Find the solutions to the following equations and draw them on the Argand diagram: (a) z 3 = (b) z 3 = (c) z = 4 (d) z = i (e) z = 4 i. J8 Show that the roots of the equation z n bz n + c = will, for general complex values of b and c and integral values of n, lie on two circles in the Argand Plane. Give a condition on b and c that these circles coincide. Find the largest value of z z when z and z are roots of z 6 z 3 + =. J9 (a) If ω 3 =, find + ω + ω. (b) If η n = for some positive integer n, state and prove the corresponding result for powers of η. *J Find the logarithms of the following: (a) (b) i (c) ( + i) (d) (x + iy) (e) ire iθ. *J A point moves in simple harmonic motion with angular frequency ω. At t = it is at a distance x from the centre of motion and is moving with a velocity v. Find the complex amplitude A in the expression Ae iωt used to represent the sinusoidal variation of distance. *J Show that the displacement from the centre of the motion in the simple harmonic motion x = 7 + 4 cos 3t + 7 sin 3t is given by the real part of the complex quantity X = (4 7i)e 3it and hence obtain the amplitude of the motion. By writing X in the form Ae iα, where A and α are real, find the time of the first two passages through the centre after t =, and the distances of the stationary points of the motion from the origin. K. Hyperbolic Functions *K Find tanh(x + y) in terms of tanh x and tanh y. *K Find the set of points in the x, y plane described by (a) (x, y) = (a cos θ, b sin θ), (b) (x, y) = (a cosh θ, b sinh θ) where a, b are real and fixed and θ varies. Find the equation of the locus and draw it in each case.
*K3 Sketch the graphs and find the indefinite integrals of each of the following functions: (a) e x (b) sinh x (c) cosh x (d) tanh x (e) log x (f) cosh x (g) tanh x Note: the last integrals need to be done by parts; eg in (e) write log x as log x. *K4 Find the integral b a dx x + x + (Hint: use a hyperbolic angle substitution). K5 Evaluate tanh x as a logarithm and differentiate it. *K6 By writing both sin x and sinh x as exponentials, calculate sin x sinh x dx. *K7 Write the integrand as the real part of a complex exponential, and deduce that e ax cos(bx)dx = 6: ORDINARY DIFFERENTIAL EQUATIONS *L Solve by separation of variables:. eax a (a cos(bx) + b sin(bx)) + const. + b (a) x 3 dx + (y + ) =. (b) dx = *L Solve by change of variable and separation: 4y x(y 3). (x + y + ) dx + (x + y + ) + x 3 =. *L3 Solve by use of integrating factors: (a) + xy = 4x. dx (b) dx + ( 3x )x 3 y =. 3
*L4 Solve by change of variables and the use of integrating factors: (a) dx y = xy5. (b) dx + y = y (cos x sin x). L5 (RHB.4/4.) Solve the homogeneous equation *L6 Solve: (a) (b) (y x) + (x + 3y) =. dx dx = y ( y ) x + tan x (ln y x) dx y ln y =. (Find a substitution which simplifies the equation.) (c) xy dx + (x + y + x) =. *L7 Solve the following differential equations subject to the boundary conditions y() =, (a) (b) (c) dx () =. d y dx 5 dx + 6y =. ( d dx + n. ) y =. ( d dx + d ) dx + 4 y =. (d) ( ) d dx + 9 y = 8. (e) ( d dx 3 d ) dx + y = e 5x. 4 Question continues on next page.
(L7, continued) For parts (f) and (g) find the general solution (f) d y dx dx + y = (g) d y dx dx + y = ex + e x. L8 For the differential equation dx = 3y 3 (a) find a general solution, (b) show that y = is a solution, (a singular solution), (c) show further that (x k ) 3, x k y = k < x < k (x k ) 3, k x is a solution, where k, k are constants. (d) How is it possible to have a solution such as (b) which is not obtainable from the general solution in (a)? L9 Find the complementary function, and a particular integral, of the equation in the case c. y ( + c)y + ( + c)y = e (+c)x, ( ) Show that there is a solution of the form y = f(x, c) where for any A, B and c (c ). f(x, c) = Ae x + B ex c (ecx ) + ex c (ecx e cx + ), Find the limit of f(x, c) as c. Hence or otherwise find the complementary function and a particular integral for ( ) in the case c =. L Show that the equation may be written as η dc dη = d C dη ( d ln dc ) = η dη dη. Hence show that if dc dη () = A and C() = B then η C(η) = B + A exp( t /4) dt. 5
*L (a) Solve the pair of differential equations dx dt = ax, dt = ay + bx where a and b are constants, subject to the initial conditions x() =, y() =. (b) Show that the pair of differential equations dx dt = x xy, dt = y + xy can be transformed into a single first order equation of the form Hence, or otherwise, show that is independent of t. 7: PARTIAL DIFFERENTIATION M For the function dx = f(x, y). e x x ey y f(x, y) = x(y + y ), find the components of the vector ( f/ x, f/ y), known as the gradient vector, at the points (, ), (, ), (, ) and (, ). Make a sketch showing the directions of the gradient vector at these points. M The period T of a simple pendulum of length l swinging in a gravitational field g is given by T = π(l/g). Estimate the percentage error in a measurement of g due to (i) a.% error in measuring l (ii) a.% error in measuring T? [Hint: take logs before differentiating!] M3 For f(x, y) = exp( xy), find ( f/ x) y and ( f/ y) x. Check that ( / x) y ( f/ y) x = ( / y) x ( f/ x) y. Find ( f/ r) θ and ( f/ θ) r, (i) using the chain rule, (ii) by first expressing f in terms of polar coordinates r, θ, and check that the two methods give the same results. M4 If xyz + x 3 + y 4 + z 5 = (an implicit equation for any of the variables x, y, z in terms of the other two), find ( ) x y z and show that their product is., ( ) y z x 6, ( ) z x y
M5 Van der Waal s equation (p + a/v )(V b) = RT is an early (and in many ways a remarkably successful) attempt to represent the relation between the pressure p, volume V and temperature T of a real gas. (R, a and b are constants for a given mass of the gas.) Calculate expressions for ( p/ V ) T, ( V/ T ) p and ( T/ p) V, and verify that their product is. M6 f(x, y) is a scalar function of position on the x y plane. Position may also be specified by Cartesian coordinates u, v which are referred to axes rotated by angle θ from the x and y axes. Show that f x + f y = f u + f v, i.e. the -dimensional operator is invariant under a rotation of axes. N. Unconstrained Stationary Values N The height h of each point (x, y) of an area of land is given by h(x, y) = a(x + y) x + y + a, where a is a positive constant. Find the locations and heights of the highest and lowest points of the terrain, and also those along the x and y axes. Sketch a map of the region by showing contours of constant h in the (x, y) plane. N (a) Find the stationary points of the function z = (x y )e x y. (b) Find the contours on which z = and examine the behaviour of z on the axes. Hence, or otherwise, determine the character of the stationary points. Sketch the contours. N3 Find and classify the stationary values of the functions (a) (x + y + ) (b) sin x sin y ( < x < π, < y < π) (c) (harder question) (xy y) exp(x x y ). 7