PHY 301: Mathematical Methods I Curvilinear Coordinate System (10-12 Lectures)

Transcription

1 PHY 301: Mathematical Methods I Curvilinear Coordinate System (10-12 Lectures) Dr. Alok Kumar Department of Physical Sciences IISER, Bhopal Abstract The Curvilinear co-ordinates are the common name of different sets of coordinates other than Cartesian coordinates. In many problems of physics and applied mathematics it is usually necessary to write vector equations in terms of suitable coordinates instead of Cartesian coordinates. First, we develop the vector analysis in rectangular Cartesian coordinate to see the fundamental role played by the vector-valued differential operator,. All objects of interests are constructed with the del operator - the gradient of a scalar field, the divergence of a vector field and the curl of a vector field. Later we generalize the results to the more general setting, orthogonal curvilinear coordinate system and it will be a matter of taking into account the scale factors h 1, h 2 and h 3. For the most general coordinate transformation we have to consider the tensor analysis. Rectangular Cartesian coordinate is a special case of the orthogonal Curvilinear coordinate system, what we mean is h 1 = h 2 = h 3 = 1. Intuitively, the scale factor is 1-dimensional version of the Jacobian and we encounter the Jacobian firstly while handling the multiple integral. Please do not take this lecture notes at the face value, verify and check everything and spot the conceptual mistakes etc. to improve it. address: alok@iiserbhopal.ac.in 1

2 The aim of vector analysis (calculus) to have the notion of the differentiation and the integration in more than one dimension. The derivative of a function in one dimension gives the rate of the change of the function with respect to an independent variable. In more than one dimension, there are infinitely many direction and the derivative of a scalar function φ(x, y, z) will depend on the chosen direction. In short, the question how fast does φ(x, y, z) vary? has an infinite number of answers, one for each direction we might choose to explore [2]. Geometrically, the scalar function φ(x, y, z) represents a family of surfaces and on a particular surface the value of the scalar function does not change. At a point P (x, y, z) on a surface, there is a unique normal direction perpendicular to the surface and the directional derivative along the normal direction is called the gradient of the scalar function, grad φ = dφ ˆn (1) dn where ˆn is the unit vector along the normal direction. This definition is given independently of any coordinate system. The gradient of a scalar function is the rate of space variation along the normal to the surface on which it remains constant [5]. With the rule of partial differentiation, dφ = ( ) φ dx + x We can rewrite the equation (2) as dφ = ( ) φ dy + y ( ) φ dz. (2) z ( φ xî + φ y ĵ + φ ) z ˆk.(dxî + dyĵ + dzˆk) (3) = ( φ).(d l). The vector φ = (î x φ + ĵ y φ + ˆk z φ) is the gradient of the scalar function φ(x, y, z). This definition of the gradient is with respect to rectangular Cartesian coordinate system. Both definitions of the gradient of a scalar functions equation (1) and equation (3) are equivalent. In equation (3) dφ is a scalar and d l is a vector and therefore φ must be a vector. Like any vector, a gradient has magnitude and direction. dφ = ( φ).(d l) (4) = φ d l cos(θ). 2

3 In equation (4), the change in dφ will be maximal for θ = π for fixed d l 2 i.e. when d l points in the direction of φ for fixed d l. Therefore we state, the gradient φ points in the direction of the maximum increase of the function φ. From equation (1), it is obvious that the direction of the gradient is that of the normal to the surface given by the φ(x, y, z) = c. This can be taken as geometrical interpretation of the gradient of a scalar function. For a point P (x, y, z) to be stationary point, φ(x, y, z) = 0. It is to be noticed that in the definition of the gradient of a scalar function φ(x, y, z) = î x φ(x, y, z) + ĵ y φ(x, y, z) + ˆk z φ(x, y, z), the formal appearance of the del operator is very crucial. Now φ is a vector quantity, φ(x, y, z) is a scalar quantity and therefore the differential operator xî + yĵ + ˆk must behave like a vector quantity (like Quotient rule). z It is not a vector in usual sense but it is a vector-valued differential operator and hungry to differentiate [1]. Anything comes behind the, get just multiplied ordinarily and anything comes in front of it get differentiated. So the mimics the behaviour of an ordinary vector by acting upon any function and differentiating it. This operator play a very significant role in the whole vector calculus. and 2 are invariant under rotation i.e. î x + ĵ y + ˆk z = î x + ĵ y + ˆk z. Just imagine, with the del operator we have the power of vector algebra in one hand and the power differential calculus in other hand. How to prove the equivalence of two definitions of the gradient of a scalar function, equation (1) and equation (3)? Let us consider the scalar function in three dimensions, φ(x, y, z) = x 2 +y 2 + z 2 = c. Obviously it represents a family of spheres centred at the (0, 0, 0) with the radius c and on a particular sphere the value of the scalar function φ(x, y, z) = x 2 + y 2 + z 2 = c remains constant i.e. c will be a constant on a particular sphere. More generally, any scalar function φ(x, y, z) = c represents a family of surfaces with different values of c. Let us consider two nearby surfaces, φ(x, y, z) = c and φ(x, y, z) = c + c with the origin chosen at the point O. P is a point on φ(x, y, z) = c and P on φ(x, y, z) = c + c. OP = r, OP = r + d r and P P = d r. Let the unit normal vector ˆn at the point P makes an angle θ with P P = d r. Hence dn = ˆn.d r = dr cos θ. Therefore, from equation (1), dφ = dφ dφ dn = ˆn.d r = grad φ.d r. (5) dn dn 3

4 We compare equation (3) and equation (5) to get grad φ = dφ ˆn = î φ dn x + ĵ φ φ + ˆk y z = φ (6) and hence the both definitions are equivalent. The vector field is defined as the set all unique vectors corresponding to each point (x, y, z) R 3. Needless to mention the pivotal role of the del operator in the study of vector calculus. What are the quantities of interest for the study the vector field? In Newtonian mechanics we study the motion of any rigid body by breaking its motion two parts : the translational motion of the centre of mass and the pure rotational motion in the centre of mass frame. Similarly we analyse the vector field - the linear content (the divergence) and the rotational content (the curl) at a point. The Divergence of a Vector Field: The divergence of a vector field is closely associated with the notion of flux of a vector field and we encounter this in the study Gauss law for the electric field flux. The flux of any vector field across any small area around a point is defined as dφ A = A.d S = A.ˆn ds (7) where ˆn the outward normal unit vector. Physically it represents the flow of the vector through the elementary area. The total flux through the whole arbitrary small closed surface around the point is given by φ A = A.dS = A.ˆn ds. (8) S The divergence of a vector field at a point of the vector field is the limiting value of the ratio of the flux of the vector field across an elementary closed surface around the point to the volume of the enclosure when the volume of the enclosure contracted on to the point. div A S A.d = lim S. (9) τ 0 τ This definition is independently of any coordinate system. Let us choose Rectangular Cartesian coordinate system and in this system the Divergence of a Vector Field will have the following computational form with the definition in equation (9) S div A =. A = A x x + A y y + A z z. (10) 4

5 Physically the divergence of a vector field at a point represents the source strength of the vector field. If the divergence at a point is positive, the point is behaving like the source, if it is negative the point is behaving like the sink. If the divergence of a vector field is zero at all points, we say the vector field is Solenoidal i.e.. A = 0 (x, y, z) R 3. Intuitively, it is associated with the outward normal components on an arbitrary surface around the point and hence it is capturing the linear contents (w.r.t. normal direction) of the vector field. The definition in equation (9) is good motivation for the Gauss Divergence Theorem. The surface integral of a vector field carried over the entire surface of a closed figure is equal to the volume integral of the divergence of the vector field. That is, A.dS =. A dτ (11) S where S is the surface area of the closed figure and τ is the volume of the space enclosed by the same figure. The Curl of a Vector Field: The curl of a vector field around a point is defined as a vector along the normal to an elementary area centered on the point and of the magnitude equal to the limiting value of the ratio of the line integral to the area itself as the area is contracted into that point. τ A.d curl A l = lim ˆn. (12) S 0 S where S is an elementary area and ˆn is the unit vector along the normal to S. This definition is also independently of any coordinate system. In the Rectangular Cartesian coordinate system the above definition will be A î ĵ ˆk =. (13) x y z A x A y A z With the definition in equation (12) we can motivate to Stokes s Theorem, This theorem states that the line integral of a vector field along the closed path is equal to the surface integral of the curl of the vector carried throughout the area bounded by the path. That is, A.d l = curla.d S. (14) S 5

6 If the curl A = 0 (x, y, z) R 3, the vector field A is called irrotational field. Physically the curl of a vector field like vortex in the fluid motion and if we put a wheel suitably in a rotational fluid it will start rotating. Laminar flow is like irrotational velocity vector field. The curl of a central field A(r) = f(r)ˆr is zero, it has no vortex like structure, what makes it a conservative field. The gradient of a central scalar function can be calculated directly, f(r) = ˆr d f(r). dr Theorem : Curl-free (or irrotational ) fields. The following conditions are equivalent for a curl-free vector field A [2]: 1. A = 0 (x, y, z) R 3 ; 2. b a A.d l is independent of the path; 3. A.d l = 0 for any closed loop; 4. A is the gradient of some non-unique scalar field, A = φ. Theorem : Divergence-less (or solenoidal ) fields. The following conditions are equivalent for a divergence-less vector field A [2]: 1.. A = 0 (x, y, z) R 3 ; 2. surface A.d a is independent of the surface, for any given boundary line; 3. surface A.d a = 0 for any closed surface; 4. A is the curl of some vector field, A = W, where W is not unique. The Laplacian operator is second order differential operator is defined as 2 =. = xx + yy + zz and satisfy the identity in three dimension, ( ) ( ) 1 2 =. r r = 4πδ(r) (15) r 3 and 1 r is interpreted as the Green s function the Laplacian operator 2. For proof of equation (15), see reference [2]. The equation is closely related to the identity in 3-dimension.(r n r) = (n + 3)r n. (16) 6

7 Left hand side of (16) is zero for n = 3, it is related to (15) in three dimension. In two dimension.(r n r) = (n + 2)r n. (17) In (17) n = 2 makes zero. So in two dimension the Green s function of the 2 is ln(r) instead of 1 r [6]. 2 (ln(r)) = kδ(r) (18) where k is the strength of the delta function to be determined. For m dimension we conjecture.(r n r) = (n + m)r n. (19) Writing 2 = div grad =. and using grad f(r) = ˆr d dr f(r) 2 (r n ) = div ( ) ˆr d(rn ) =. dr ( n rr n 2) = n(n + m 2)r n 2 (20) for the dimension, m. There are many vector identities see [2, 3, 4] but two are of utmost importance, 1. The curl of gradient of a scalar function is zero. φ(x, y, z) = 0. Intuitively it state that the rotational content of a gradient (conservative) vector field zero. 2. The divergence of curl of a vector field is zero..( A) = 0. Intuitively it state that the curl of a vector field has no linear content and hence its divergence is zero. Helmholtz s Theorem 1: A vector field is uniquely specified by giving its divergence and its curl within a simply connected region (without holes) and its normal component over the boundary. Helmholtz s Theorem 2: A vector field satisfying. A = s and A = c with both the source and the circulation densities vanishing at infinity may be written as the sum of two parts, one of which is irrotational, the other of which is solenoidal.[4] We developed vector analysis in the rectangular Cartesian coordinate system. A Cartesian coordinate system offers the unique advantage that all three unit 7

8 vectors, î, ĵ and ˆk are constants in magnitude as well as in direction. Unfortunately, not all physical problems are well adapted to a solution in Cartesian coordinates. For instance, if we have a central force problem, F = ˆrF (r), such as gravitational force, Cartesian coordinates may be unusually inappropriate. Such a problem demands the use of a coordinate system in which the radial distance is taken to be one of the coordinates, that is, spherical polar coordinates. The point is that the coordinate is chosen to fit the problem, to exploit any constraint or symmetry present in it. Then it is likely to be more readily soluble than if we had forced it into a Cartesian framework [4]. We generalise the results developed in the rectangular Cartesian coordinate system to orthogonal Curvilinear coordinate system and we see the translation of all results developed so far will be boiled down to taking into account the scale factors h 1, h 2, h 3. We follow reference [3] for Curvilinear formulation. Our physical three dimensional space is the continuum of physically points. In Cartesian it is represented by the set {(x, y, z) R 3 }. The same physical point is denoted by (u 1, u 2, u 3 ) in a general coordinate system. Both (x, y, z) and (u 1, u 2, u 3 ) represent the same physical point P and therefore we demand both coordinates are related via some mapping. x = x(u 1, u 2, u 3 ) y = y(u 1, u 2, u 3 ) z = z(u 1, u 2, u 3 ) u 1 = u 1 (x, y, z) u 2 = u 2 (x, y, z) u 3 = u 3 (x, y, z). (21) We demand that the function between two sets (x, y, z) and (u 1, u 2, u 3 ) should be single valued and have continuous derivatives so that the correspondence is unique. Therefore given a physical point P with rectangular coordinates (x, y, z) we can associate a unique set of coordinates (u 1, u 2, u 3 ) called the curvilinear coordinates. The set of the transformation (21) define a transformation of coordinates. Now there are three coordinate surfaces and three coordinate curves given by u 1 = c 1 u 2 = c 2 u 3 = c 3 u 1 = c 1 &u 2 = c 2 coordinate surfaces; u 1 = c 1 &u 3 = c 3 coordinate curves. u 2 = c 2 &u 3 = c 3 (22) see the Fig. 1 in chapter 7 of reference [3]. If we keep one of the coordinate fixed and let other two vary, a surface of definite shape is traced. Such surfaces are called coordinate surfaces. Two surfaces cut to give a curved line, is called coordinate curve. Let r = îx + ĵy + ˆkz be the position vector of 8

9 a physical point P and its position vector in terms of curvilinear coordinate is r = r(u 1, u 2, u 3 ). r(u 1,u 2,u 3 ), r(u 1,u 2,u 3 ) and r(u 1,u 2,u 3 ) are the tangent vectors along the u 1, u 2 and u 3 curves respectively at the point P. The unit tangent vectors (ê 1, ê 2, ê 3 ) are h 1 = r(u 1,u 2,u 3 ) h 1 ê 1 = r(u 1,u 2,u 3 ) h 2 ê 2 = r(u 1,u 2,u 3 ) h 3 ê 3 = r(u 1,u 2,u 3 ) h 2 = r(u 1,u 2,u 3 ) h 3 = r(u 1,u 2,u 3 ) scale f actors (23) where h 1, h 2 and h 3 are called scale factors. We know that the gradient u 1 at the point P is along the normal to the surface u 1 = c 1 and similarly u 2 and u 3 are along the normal to the surfaces u 2 = c 2 and u 3 = c 3. The unit normal vectors at the point P u Ê 1 = 1 u 1 u Ê 2 = 2 u. (24) 2 u Ê 3 = 3 u 3 It is a simple exercise to show that two bases ( r(u 1,u 2,u 3 ), r(u 1,u 2,u 3 ), r(u ) 1,u 2,u 3 ) and ( u 1, u 2, u 3 ) are dual (reciprocal) to each other, see the solved problem 15 of reference [3]. Any vector A can be written as A = A i ê i = a i Ê i (25) where dummies indices means sum over is understood. For orthogonal curvilinear coordinate, e 1.e 2 = e 2.e 3 = e 1.e 3 = 0 and we are interested in this system. The most fundamental quantity in geometry is the distance between two neighbour points ds 2 = dx 2 + dy 2 + dz 3 = d r.d r. (26) From r = r(u 1, u 2, u 3 ) with the rule of partial differentiation d r = r du 1 + r du 2 + r du 3 = h 1 du 1 ê 1 + h 2 du 2 ê 2 + h 3 du 3 ê 3. (27) The equation (26) can written as ds 2 = d r.d r = (h 1 du 1 ê 1 + h 2 du 2 ê 2 + h 3 du 3 ê 3 ).(h 1 du 1 ê 1 + h 2 du 2 ê 2 + h 3 du 3 ê 3 ) (28) 9

10 ds 2 = ( ) du 1 du 2 du 3 h 2 1ê 1.ê 1 h 1 h 2 ê 1.ê 2 h 1 h 3 ê 1.ê 3 h 2 h 1 ê 1.ê 2 h 2 2ê 2.ê 2 h 2 h 3 ê 2.ê 3 h 1 h 3 ê 1.ê 3 h 3 h 2 ê 2.ê 3 h 2 3ê 3.ê 3 du 1 du 2 du 3 (29) for orthonormal system the off-diagonal elements of the equation (29) will be zero. Therefore in the orthogonal curvilinear system the distance element (the first fundamental form ) is given ds 2 = h 2 1(du 1 ) 2 + h 2 2(du 2 ) 2 + h 2 3(du 3 ) 2. (30) Comparing the Cartesian distance element as in equation (26) with the orthogonal curvilinear distance element as in equation (30), we find the change along u 1 is by the scale factor h 1 and similar story for other coordinate curve. The volume element would be given equation (34), scalar triple product. The gradient, the divergence, the curl and the Laplacian in the (u 1, u 2, u 3 ) is a matter taking into account the scale factor properly and the proof of all equations (36-39) are given in the Spiegel [3]. We need to study two most commonly used Orthogonal Curvilinear coordinate systems in full details. 1. Cylindrical Coordinates (ρ, φ, z) :The following coordinate transformation is called Cylindrical Coordinates. x = ρ cos φ, y = ρ sin φ, z = z; where ρ 0, 0 φ 2π, < z < h ρ = 1, h φ = ρ, h z = Spherical Coordinates (r, θ, φ) :The following coordinate transformation is called Spherical Coordinates. x = r sin θ cos φ, y = r sin θ sin φ, z = r cos θ; where r 0, 0 φ 2π, 0 θ π h ρ = 1, h θ = r, h φ = r sin θ. It is to noted that for any orthogonal curvilinear coordinate system what is important the scale factors as we can see in equations (36-39). 10

11 References 1. Richard P. Feynman, Robert B. Leighton and Matthew Sands Lectures on Physics, Volume 2 (Indian edition), Narosa Publishing House, (986).e 2. David J. Griffiths, Introduction to Electrodynamics (2nd edition), Prentice-Hall of India Private Limited, (1998). 3. Murray R. Spiegel, Theory and Problems of Vector Analysis (SI metric edition), Schaum s Outline Series, (1974). 4. George B. Arfken and Hans J. Weber, Mathematical Methods for Physicists (sixth edition), Academic Press, (2005). 5. N. N. Ghosh, Teach Yourself Physics Mathematical Physics, Bharati Bhawan (Patna, Bihar), (1990). 6. B. S. Agarwal 1, Mechanics, Kedar Nath Ram Nath (Meerut, U.P.), (2000). 1 All titles by him highly-recommended for undergraduate students. 11

12 Quick to Curvilinear Coordinate System Given a physical point P is represented by x i in rectangular Cartesian and by u i in any general coordinate system. u i = u i (x j ). (31) In R 3 two natural bases dual to each other are ( r(u 1,u 2,u 3 ), r(u 1,u 2,u 3 ), r(u 1,u 2,u 3 ) ) and ( u 1, u 2, u 3 ). ds 2 = d r.d r (32) d r(u 1, u 2, u 3 ) = r(u 1, u 2, u 3 ) du 1 + r(u 1, u 2, u 3 ) du 2 + r(u 1, u 2, u 3 ) du 3 (33) dv = (h 1 du 1 ê 1 ).((h 2 du 2 ê 2 ) (h 3 du 3 ê 3 )) (34) h 1 = h 2 = h 3 = u i = êi h i r(u 1, u 2, u 3 ) r(u 1, u 2, u 3 ) r(u 1, u 2, u 3 ) ; h 1ê 1 = r(u 1, u 2, u 3 ) ; h 2ê 2 = r(u 1, u 2, u 3 ) ; h 3ê 3 = r(u 1, u 2, u 3 ) (35) u i = 1 h i Ê i = u i u i ê i = Êi f or orthogonal system. h 1, h 2 and h 3 are scale factors owing to curvilinear nature of the coordinate system. φ = 1 h 1 φ ê h 2 φ ê h 3 φ ê 3 (36) 12

13 Table 1: Common Scale Factors Curvilinear Cartesian Spherical Cylindrical u 1 x r ρ u 2 y θ φ u 3 z φ z h h 2 1 r ρ h 3 1 r sin θ 1. A = A = 2 φ = [ (A1 h 2 h 3 ) + (A 2h 1 h 3 ) + (A 3h 2 h 1 ) 1 h 1 h 2 h 3 ê 1 ê 2 ê h 1 h 2 h 3 A 1 A 2 A 3 [ ( ) 1 h2 h 3 φ h 1 h 2 h 3 h 1 + ( h1 h 3 h 2 ) φ ] + ( h1 h 2 h 3 (37) (38) )] φ (39) where (ê 1, ê 2, ê 3 ) is a curvilinear orthonormal basis. d A = (.d r) A (40) Extensions of the above results are achieved by a more general theory of curvilinear systems using the methods of tensor analysis. All these results are smoothly derived in Spiegel [3]. 13

14 Suggested Assignments: 1. If A and B are irrotational, prove that A B is solenoidal. 2. If f(r) is differentiable, prove that f(r) r is irrotational. 3. If U and V are differentiable scalar fields, prove that U V is solenoidal. 4. Show that A = (2x 2 + 8xy 2 z)î + (3x 2 y 3xy)ĵ (4y 2 z 2 + 2x 3 z)ˆk is not solenoidal but B = xyz 2 A is solenoidal. 5. In what direction from the point (1,2,3) is the directional derivative of Φ = 2xz y 2 a maximum? What is the magnitude of this maximum? 6. Find the values of the constants a, b, c so that the directional derivative of Φ = axy 2 + byz + cz 2 x 3 at (1,2,-1) has maximum magnitude 64 in a direction parallel to the z-axis. 7. Verify Stokes theorem for A = (y z + 2)î + (yz + 4)ĵ xzˆk, where S is the surface of the cube x=0, y=0,z=0,x=2,y=2,z=2 above xy plane. 8. Verify Green s theorem in the plane for C (3x2 8y 2 )dx+(4y 6xy)dy, where C is the boundary of the region defined by :y = x, y = x Prove that φ(x, y, z) = x 2 + y 2 + z 2 is scalar invariant under rotation of axes. 10. Show that under a rotation = î x + ĵ y + ˆk z = î x + ĵ y + ˆk = z (41) 11. Show that under a rotation Laplacian operator is invariant. 12. Given the dyadic φ = îî+ĵĵ + ˆkˆk, evaluate r.( φ. r) and ( φ. r). r. Is there any ambiguity in writing r. φ. r. What is the geometrical significance of r. φ. r = If A = xzî y 2 ĵ + yz 2ˆk and B = 2z 2 î xyĵ + y 3ˆk, give a possible significance to ( A ) B at the point (1,-1,1). 14

15 14. Verify the divergence theorem for A = 4xî 2y 2 ĵ + z 2ˆk taken over the region bounded by x 2 + y 2 = 4, z = 0 and z = Show that 1. ( 1 r ) = r r 3 2..( r) = 3 3. ( r) = ( a r) = 2 a where a is a constant vector. 16. Problem numbers 37, 42, 46, 50, 53, 54, 55, 58, 59, 60, 67 of chapter 7 of reference [3]. It is always fine to solve many more problems as in reference [2, 3, 4]. With the best of luck!! 15