Mathematical Methods 2 (PART I) Lecturer: Dr. D.J. Miller Room 535, Kelvin Building [email protected] Location: Room 312, Kelvin Building Recommended Text: Boas, 3 rd Edition 1 Outline 1. Curvilinear coordinate systems 2. Partial Differential Equations in cylindrical and spherical polar coordinates 3. Legendre Polynomials and Bessel Functions 4. Hermite and Laguerre Polynomials 5. Gamma and Beta Functions, and Stirling’s approximation 2 1. Curvilinear coordinate systems 1.1 Revision of vector calculus with Cartesian coordinates Consider a 3-dimensional vector space. Any point in the space can be written in terms of three coordinates ___________ and three basis vectors Sometimes the coordinates are written as _______ and the unit vectors as (I will use both notations, but avoid i, j and k). Any vector A in this space, can be written in terms of these unit vectors as, I will sometimes assume the Einstein Summation convention and omit the summation sign Σ when I have two repeated indices (unless otherwise specified), e.g. 3 These unit vectors are orthonormal (orthogonal and of unit length) if Kronecker delta The position vector is given by remember Einstein r summation convention! The scalar product of two vectors A and B is The vector product of two vectors A and B is this has a sum over i, j and k where is the Levi Civita symbol , which is +1 for an even permutation of 1, 2 ǫijk and 3 and -1 for an odd permutation, and zero if any of the indices are the same, e.g. 4 For example, the z-component is: The gradient operator ∇ is given by From this, and the definitions above, follow the gradient of a scalar field φ(r), one number for each value of r and the divergence and curl of a vector field A(r), three numbers for each value of r 5 Exercise : Using the Einstein summation convention, prove (you will need for the this one), and 6 1.2 Curvilinear coordinates in physics problems Key Point: Understand the importance of curvilinear coordinates in solving physics problems Very often, physical systems are described by classical (or quantum) field theories. We have a field, e.g. , which obeys a differential equation over all space, e.g. the Schrödinger Equation The field is constrained by the differential equation and by boundary conditions . The differential equation is independent of coordinate system (though may look very different in different coordinate systems) The boundary condition is often very much simpler in one particular choice of coordinates, making the solution simpler in these coordinates. Use symmetries of the boundary conditions (incl. the position of sources/sinks) to choose your coordinate system. 7 Field Differential Equation General Solution scenario coordinate independent system Boundary Specific Conditions Solution scenario dependent 8 Example: Spherical Polar Coordinates Describe the position of a point in space using the distance from the origin, r, and two angles, θ and φ. This is useful in problems with spherical symmetry . e.g. Consider a hollow sphere or radius R kept at a constant temperature T. The temperature inside the sphere obeys Laplace’s equation, which is coordinate system independent. However, the boundary condition is much more easily expressed in spherical polar coordinates: 9 Example: Cylindrical Coordinates Describe the position of a point in space using the distance along the z-axis, the distance from the z- axis, r, and an angle, θ. This is useful in problems with cylindrical symmetry . e.g. Consider a wire with a uniform charge density ρ0. The electric potential generated by the charge distribution is given by Poisson’s equation, which is coordinate system independent (and becomes Laplace’s equation away from the charge). However, the charge density is much more easily expressed in cylindrical polar coordinates (with the wire running along the z-axis): 10 1.3 Unit vectors and scale factors Key Point: Write down the definition of basis vectors and scale factors for general curvilinear coordinates In Cartesian coordinates, if I change the x-coordinate of the position vector r by an amount dx , then the object has moved a distance dx . If I change all of the coordinates (infinitesimally) at once, I move r a distance dr where using Pythagoras’ theorem. dr is know as the infinitesimal line element . This may seem obvious, but the analogue is not true in general curvilinear coordinates . 11 e.g. Imagine changing θ by and amount d θ in cylindrical coordinates: The distance moved is for dθ small. (The z-direction is out of the slide.) How far we move depends on the value of the coordinate r. How far we move for a particular coordinate shift is known as a scale factor . Also notice that the direction we move depends on the coordinate θ. The direction we move for a particular coordinate shift is given by the basis vector . 12 Imagine a general coordinate system described by ( =1,2,3) and related to Cartesians qi i via As we change the coordinate , the position vector will move: qi r (no sum over i) basis vector scale factor The scale factor is the magnitude of the vector , i.e. The basis vector is the unit vector in the direction of , i.e. (no sum) 13 As before, the basis vectors are orthogonal if Let’s call the distance moved by changing the coordinate by an amount ds i qi dq i (no sum) For orthogonal coordinates, we can use Pythogoras’ theorem again to write the total displacement from altering all three coordinates: More formally, the change in the position vector is So, 14 More generally, one may define the metric of the space according to gij , metric (no sum) So, in our case we have , but this can in principle be more complicated. The volume element is given by ds ds ds 15 Key Point: Derive the scale factors and unit vectors for various coordinate systems (no sum) Example : Cartesian Coordinates (trivial!) The position vector is So, These are already unit vectors, so are the basis vectors and the scale factors are, The volume element is , and the square of the infinitesimal line element is . 16 Example : Spherical Polar Coordinates They are related to Cartesians by: and the position vector is ( but since changes with and , it is more convenient to use , and .) er θ φ ex ey ez So, 17 Normalize these vectors to find the basis vectors and scale factors: So, we have scale factors : , , . basis vectors : The volume element is , and the square of the infinitesimal line element is . 18 We could have derived the scale factors directly from the square of the infinitesimal line element: So, write ds =dx +dy +dz and plug in dx , dy and dz (above). The cross terms vanish and we are left with The lack of cross-terms tells us the coordinates are orthogonal, and the coefficients of the diagonal terms are the square of the scale factors. Exercise : Find the scale factors, unit vectors, volume element and infinitesimal line element-squared for cylindrical coordinates. 19 1.3 The Gradient Key Point: Derive an expression for the gradient in general orthogonal curvilinear co-ordinate systems The gradient in Cartesian coordinates was: This tells us how fast a field changes with position, i.e. with respect to the distance moved NOT how much the parameter changes ⇒ We must take into account the scale factors. For a curvilinear coordinates , recall the distance moved for a change was qi dq i ds = h dq , so the component of the gradient in the direction of e is i qi i qi (no sum) and the gradient is 20 1.5 The Divergence, Curl and Laplacian Operators Key Point: Derive expressions for div, grad, curl and the Laplacian in general orthogonal curvilinear co-ordinate systems Divergence We may now use the form of the gradient to derive the divergence ∇ · A changes with coordinates, so eq wej need to differentiate it too! Before going any further we need to work out ∇ · e qj Back to our definition of the gradient, acting on : qj 21 Our unit vectors form a right-handed set, so But the divergence of the left-hand-side of this equation is zero: I can repeat this argument for cyclic permutations This allows us to rearrange our equation in a nicer way. 22 these have no divergence so can be pulled in front of the ∇ But , so finally we have, 23 Curl Curl works in much the same way. We may use the result we obtained earlier: since, again, the left-hand-side is zero. Remember that so I will pick up contributions for from the terms linked by the arrows. 24 Similarly for the other terms. We can write this as a determinant: 25 Laplacian The laplacian is ∇·∇ = ∇2, so we can fairly easily insert our expression for the gradient into the expression we derived for the divergence: but So, remember both of these derivatives are acting on everything to the right (despite the brackets) 26 Key Point: Apply these expressions to various coordinate systems, including spherical polar coordinates and cylindrical coordinates. Example : Spherical Polar Coordinates: ( , , ) 27 Exercise : Find the gradient, divergence, curl and Laplacian for cylindrical coordinates. Exercise : Using spherical polar coordinates, show that and apply these results to show 28 2. Partial Differential Equations in cylindrical and spherical polar coordinates 2.1 Common Differential Equations Key Point: Write down Laplace’s equation, Poisson’s equation, the Diffusion equation, the Wave equation, the Helmholtz equation and Schrödinger’s equation. We have already seen that physical systems are often described by scalar and vector fields in space and time, that obey particular differential equations involving the operators of the previous section.