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MA156 - Mathematical Methods for Physical Sciences Conservative vector fields Contents 1 Introduction 2 2 Gradient and directional derivative 2.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2 4 3 Conservative fields 3.1 Definition . . . . . . . . . . . . . . . 3.2 Properties of conservative fields . . . 3.3 Potentials of conservative vector fields 3.3.1 Integral method . . . . . . . . 3.3.2 Differential method . . . . . . . . . . . 5 5 6 9 9 9 . . . . 10 10 11 12 14 . . . . . 4 Stokes theorem 4.1 Introduction . . . . . . . . . . . . . . . 4.2 The curl of a vector field . . . . . . . . 4.3 Stokes theorem . . . . . . . . . . . . . 4.4 Stokes theorem and conservative fields. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Divergence and Divergence theorem 15 6 Physical Applications of the divergence theorem 6.1 Divergence and the sources of vector fields . . . . . . . . . . . . . . . . . . . . . . 6.2 Divergence and Conservation laws . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Divergence and electrostatic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 16 16 18 MA156 - Conservative vector fields 2 1 Introduction Vector fields are extremely useful to describe forces. Some of these, like the gravitational or the electrostatic force, have a very important property: it is possible to associate an energy to them. A stone held on the top of a mountain has a certain gravitational potential energy that can be transformed into kinetic energy by letting it drop. It is clear that the vector field that describes the gravitational force is somewhat peculiar, it must have some additional properties with respect to a generic vector field. The purpose of these notes is to explain very briefly what are the characteristics that differentiate the gravitational field from a generic vector field to which it is not possible to associate a potential energy. In order to do this we must introduce an approach to vector fields that is the “complement” of what we have done until now. To understand this point consider, for the moment, a real function of one real variable, f (x), with a ≤ x ≤ b. The study of the properties of this function can be carried out in two ways: the integral of a function f (x) over an interval a ≤ x ≤ b requires the knowledge of the function over the entire interval and gives global properties of the function, for example its average value. The derivative of a function, instead, requires only a local knowledge of the function and gives only local information: knowing the first derivative of a function at a point x0 allows us to approximate the function in a small neighbourhood of x0 but does not gives us any information on the values of the function away from that point. The global (integration) and the local (differentiation) approaches are not unrelated: as a matter of fact the fundamental theorem of calculus states that the derivative is (very roughly) the “inverse” of the integral. The same two approaches can be used to study vector fields. Until now we have used the global approach and we have defined the line and surface integrals of vector fields. In order to study the properties of conservative vector fields we must also introduce the local approach and try to define differentiation operations that can either produce a vector field by acting on a scalar function f (x, y, z) (such operation is called the gradient) or that act directly on vector fields (these two operations are called the curl and the divergence of the vector field). We are now going to introduce the gradient of a scalar function and use it to define a special class of vector fields called conservative vector fields. We then show that these fields possess all the properties of the gravitational force field, namely that it is possible to associate a potential energy to them. Finally we will use the two other differential operations on vector fields, the curl and the divergence, to find some easy methods to identify whether a vector field is or is not conservative and to write conservation laws as partial differential equations. 2 Gradient and directional derivative 2.1 Definition Given a function of two or more variables, f (x, y) for example, we define the directional derivative of f at the point x0 = (x0 , y0 ) in the direction of the unit vector n as f (x0 + δn) − f (x0 ) ∂f = lim . ∂n δ→0 δ The geometrical interpretation of the directional derivative is that it represents the slope of the graph of f (x, y) when we move from the point (x0 , y0 ) in the direction indicated by the vector n (see Figure 1). Example Evaluate the directional derivative of f (x, y) = sin(x + y 2 ) in the direction of v = i + 2j at (0, 0). MA156 - Conservative vector fields 3 z f(x,y) y n x f Figure 1: The directional derivative in the direction of the unit vector n is the slope of the graph of the function in the direction of n. The gradient of f , ∇f , points in the direction of maximum slope. The line orthogonal to the gradient is a level line of the graph. In order to compute the directional derivative we need a unit vector, n= 1 v = √ (i + 2j) . |v| 5 The partial derivative of f (x, y) at the origin in the direction of n is √ √ f [(0, 0) + δn] − f (0, 0) f (δ/ 5, 2δ/ 5) − f (0, 0) ∂f = lim = lim = δ→0 ∂n δ→0 δ δ √ 1 sin(δ/ 5 + 4/5δ2 ) − 0 lim =√ . δ→0 δ 5 The geometrical interpretation of the directional derivative suggests that to describe the derivative of a function of two or more variables we need two pieces of information: the slope of the graph and the direction along which this slope is measured. In other words, a complete description of the derivative of a function of two or more variables entails the use of a vector. It turns out that a most sensible and useful vector is the gradient of the function defined as ∇f = ∂f ∂f ∂f i+ j+ k ∂x ∂y ∂z (1) for a function f (x, y, z). The symbol ∇ is called grad or nabla and you can think of it as a vector operator ∇= ∂ ∂ ∂ i+ j+ k ∂x ∂y ∂z that acts on the function f (x, y, z) to give Equation (1). MA156 - Conservative vector fields 4 Example - Evaluate ∇ sin(x + y 2 ). ∇ sin(x + y 2 ) = ∂ ∂ sin(x + y 2 )i + sin(x + y 2 )j = cos(x + y 2 )i + 2y cos(x + y 2 )j. ∂x ∂y 2.2 Properties The knowledge of the gradient of a function allows us to compute all the directional derivatives in a straightforward manner. Provided that the function f is differentiable then the directional derivative of f in the direction of the unit vector n is given by ∂f = ∇f · n, ∂n |n| = 1. (2) This relation allows us to find the geometrical meaning of the gradient of a function: it points in the direction of steepest ascent. To show this we can use (2) to write the directional derivative of a function f as ∂f = ∇f · n = |∇f | |n| cos(θ) = |∇f | cos(θ), ∂n where θ is the angle between the gradient and n. The directional derivative, i.e. the slope, is maximal for θ = 0, i.e. when n is parallel to ∇f . On the contrary the surface is level in the directions orthogonal to the gradient: if n is orthogonal to ∇f then n · ∇f = 0 =⇒ ∂f = 0. ∂n To summarise, the knowledge of the gradient allows us to know all the directional derivatives, the direction of steepest ascent and the level lines of the function. Exercise 1 - Show that ∇(f g) = f ∇g + g∇f . Exercise p 2 - Show that the gradient of a function of the radial distance from the origin, f (r), with r = x2 + y 2 + z 2 , is equal to ∇f (r) = df r̂, dr where r̂ is a unit vector in the direction of the radius, xi + yj + zk . r̂ = p x2 + y 2 + z 2 Use this result to show that the gradient of the function φ(r) = 1/r is equal to ∇φ = − r̂ . r2 MA156 - Conservative vector fields 5 3 Conservative fields 3.1 Definition If φ(x, y, z) is a differentiable function defined in a domain D it is possible to evaluate its gradient at every point of the domain D. The gradient of φ, ∇φ is a vector, function of the coordinates (x, y, z). In other words it is a vector field. More formally: Definition Let φ(x, y, z) be a differentiable function in a domain D. The vector field F (x, y, z) defined by F = −∇φ(x, y, z) is called a conservative vector field in D. The function φ is called a potential for F in D. Remark - We will see that a conservative vector field can be used to represent a force with an energy associated to it. The potential φ of the vector field is the potential energy of the force. The minus sign used in the definition of conservative field is just a matter of convention: this choice of sign implies that a body moves under the action of the force from a point of high potential to a point with a lower value of the potential function. Example The gravitational force field of a point mass, F =− is conservative since we can write km r̂, r2 1 . F = −∇ − r Remark - There are infinitely many potentials that correspond to the same vector field, but they all differ by a constant. Both φ1 (x, y, z) and φ2 ≡ φ1 + C, where C is a constant, produce the same vector field: −∇φ1 = −∇φ2 = F . This is consistent with the physical interpretation of the potential as the potential energy of a force: the only physically important quantity is not the energy associated to a point, but the energy difference between different points: this is independent of the value of the arbitrary constant, φ2 (r 1 ) − φ2 (r2 ) = [φ1 (r 1 ) + C] − [φ1 (r 2 ) + C] = φ1 (r1 ) − φ1 (r2 ). The following theorem completes the analogy between conservative vector fields and forces with a potential energy. We know form physics that the work done by the gravitational force on a body that moves from a point with height h1 to a point with height h2 is independent of the path taken to go from one to the other. Another way of stating this property is that in the absence of friction a ball rolling down a slide will climb up to exactly the same height it started from (see Figure 2). The mathematical equivalent of these statements is that the line integral of a conservative vector field is independent of the path and depends only on the end points. Theorem Let F be a conservative vector field on a domain D and let φ be a potential for it, F = −∇φ. Let A and B be any two points in D and let c be any curve in D that joins them. Then Z F · ds = φ(A) − φ(B). c MA156 - Conservative vector fields 6 1111111111111111111111111 0000000000000000000000000 0000000000000000000000000 1111111111111111111111111 0000000000000000000000000 1111111111111111111111111 0000000000000000000000000 1111111111111111111111111 0000000000000000000000000 1111111111111111111111111 0000000000000000000000000 1111111111111111111111111 0000000000000000000000000 1111111111111111111111111 0000000000000000000000000 1111111111111111111111111 v 0000000000000000000000000 1111111111111111111111111 0000000000000000000000000 1111111111111111111111111 0000000000000000000000000 1111111111111111111111111 0000000000000000000000000 1111111111111111111111111 0000000000000000000000000 1111111111111111111111111 0000000000000000000000000 1111111111111111111111111 h Figure 2: In the absence of friction the ball reaches the same height it started from and its energy is continually transformed from potential energy to kinetic energy in the downward part of the slope and from kinetic energy to potential energy in the upward part of it. Example - The function φ(x, y) = −xy is the potential of the conservative vector field F = −∇φ = yi + xj.√ Call √ r(t) the arc of circle of radius 1 centred at the origin that joins A = (1, 0) with B = (1/ 2, 1/ 2): r(t) = cos(t)i + sin(t)j, 0 ≤ t ≤ π/4. The line integral of F along the path r is Z π/4 Z π/4 Z dr 1 F · ds = F [r(t)] · cos(2t) dt = . dt = dt 2 r 0 0 The difference of potentials between starting and ending point is Z 1 1 1 F · ds = φ(A) − φ(B) = φ(0, 0) − φ √ , √ = . 2 2 2 r 3.2 Properties of conservative fields Given a potential it is straightforward to construct a conservative field: we just need to evaluate the gradient of the potential. The reverse problem, given a vector field can we ascertain whether it is conservative and, if so, what is its potential, is more involved. In order to solve it we need to discuss some more the properties of conservative fields. It is fairly straightforward to say if a field is not conservative. In fact, suppose that F (x, y) = F1 (x, y)i + F2 (x, y)j is a two dimensional conservative vector field with potential φ: ∂F1 ∂φ ∂2φ F = − , =− , 1 ∂x ∂y ∂y∂x =⇒ F = −∇φ =⇒ ∂φ ∂2φ ∂F2 F2 = − , =− . ∂y ∂x ∂x∂y The mixed derivatives of φ are identical, ∂2φ ∂F1 ∂F2 ∂2φ = =⇒ = . ∂y∂x ∂x∂y ∂y ∂x (3) Therefore, if F (x, y) is a conservative two dimensional vector field then (3) must hold. The inverse of this statement tells us that a field F does not satisfy relation (3) then it cannot be conservative. MA156 - Conservative vector fields y 1111111111111 0000000000000 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 P 7 y Q 1111111111111 0000000000000 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 x y 111111111111 000000000000 000000000000 111111111111 000000000000 111111111111 000000000000 111111111111 000000000000 111111111111 000000000000 111111111111 000000 111111 000000000000 111111111111 000000 111111 000000000000 111111111111 000000 111111 000000000000 111111111111 000000 111111 000000000000 111111111111 000000 111111 000000000000 111111111111 000000 111111 000000000000 111111111111 000000 111111 000000000000 111111111111 000000 111111 000000000000 111111111111 000000 111111 000000000000 111111111111 000000 111111 000000000000 111111111111 000000 111111 000000000000 111111111111 000000 111111 000000000000 111111111111 000000 111111 000000000000 111111111111 000000 111111 000000000000 111111111111 000000 111111 000000000000 111111111111 000000 111111 000000000000 111111111111 000000 111111 000000000000 111111111111 000000000000 111111111111 000000000000 111111111111 000000000000 111111111111 000000000000 111111111111 000000000000 111111111111 000000000000 111111111111 000000000000 111111111111 000000000000 111111111111 P Q x x Figure 3: Three examples of two dimensional domains: connected (left), simply connected (centre) and not connected (right). However, this condition is necessary, but not a sufficient: a vector field can satisfy relation (3) and still not be conservative. Example - Verify that the Biot-Savart magnetic field (the field generated by an infinitely long straight wire), B(x, y) = −yi + xj , x2 + y 2 (x, y) 6= (0, 0), (4) does satisfy relation (3). However, this field, like all magnetic fields, is not conservative. Remark The generalisation of (3) to three dimensions is ∂F2 ∂F1 = , ∂y ∂x ∂F1 ∂F3 = , ∂z ∂x ∂F2 ∂F3 = . ∂z ∂y (5) To characterise completely the properties of conservative fields and, ultimately, to find out how to generalise relations (3) and (5) so that they can give a sufficient criterion to identify a conservative vector field, we must characterise better the domains over which the fields are defined. We will then see that the Biot-Savart field, Equation (4) is not conservative because the domain on which it is defined, the entire (x, y)-plane minus the origin, is not “good enough”. We distinguish three types of domains (see Figure 3): 1. Connected domains - A domain D is connected if for every pair of points P and Q in D there exists a piecewise smooth curve in D from P to Q. In two dimensions a connected domain is “made up” of one piece with or without holes. 2. Simply connected domains - A connected domains D is said to be simply connected if every closed curve in D can be shrunk to a point in D without any part passing out of D. In two dimensions a domain is simply connected if it is “made up” of one piece with no holes in it. 3. Not connected domains - All domains that are not connected. In general, in a simply connected domain (a) any closed curve in D is the boundary of a surface lying wholly in D. (b) If c1 and c2 are two curves in D having the same end points, then c1 can be deformed continuously into c2 remaining in D throughout the entire deformation process (see Figure 4). MA156 - Conservative vector fields 8 y B c2 c1 A x Figure 4: In a simply connected domain any curve c1 can be smoothly deformed into another curve c2 with the same end points. We are now in a position to characterise in a compete manner the properties of conservative vector fields: Theorem - Let D be an open connected domain and let F be a smooth vector field defined in D. The following statements are equivalent: (a) F is conservative in D; R (b) c F · ds = 0 for every piecewise smooth, closed curve in D; R (c) Given any two points A and B in D, the line integral of F , c F · ds has the same value for all piecewise smooth curves in D starting at A and ending at B. Remark 1 - This theorem is the mathematical formulation of some well known properties of forces with a potential energy. Point (b) states that the gravitational force makes no total work if the starting point of the trajectory is the same as the end point. Points (c), instead, states that the work done by the gravitational force is independent of the particle trajectory, but depends only on the starting and ending point. Remark 2 - This theorem shows why the Biot-Savart field, Equation (4), is not conservative. Its integral around a circle of radius a centred at the origin, r(t) = a cos(t)i + a sin(t)j, 0 ≤ t ≤ 2π, is Z 2π Z −a sin(t)i + a cos(t)j B · ds = · [−a sin(t)i + a cos(t)j] dt = 2π. a2 r 0 This theorem is a partial answer to the question: given a vector field how can we know whether it is conservative? Unfortunately it is not a very simple answer. In order to state whether a field is conservative we need, for example, to evaluate its line integral on all closed paths contained in D and verify that it is always zero. We need to find a simpler test to check whether a given field is or is not conservative. To do this we need to introduce another derivative, the curl of a vector field and use Stokes theorem. Before doing this, however, we will discuss how to find the potential of a given conservative vector field. MA156 - Conservative vector fields 9 3.3 Potentials of conservative vector fields Suppose that the field F (x, y) = F1 (x, y)i + F2 (x, y)j is conservative. There are two methods to find its potential, i.e. a function φ(x, y) such that ∇φ = −F . 3.3.1 Integral method We can use the theorem that say that that the line integral of a conservative vector field on any path c joining two points P0 = (x0 , y0 ) and P1 = (x1 , y1 ) is equal to the difference of the potentials between the two points, Z F · ds = φ(x0 , y0 ) − φ(x1 , y1 ), c to define the potential function. Choose a reference point P0 = (x0 , y0 ) and assign the value of the potential at this point, φ(x0 , y0 ) = C. The value of the potential at any point P1 = (x1 , y1 ) is Z Z φ(x1 , y1 ) = F · ds + φ(x0 , y0 ) = F · ds + C, c (6) c where the line integral is evaluated on any path c that joins P1 with the reference point P0 . Equation (6) defines the function φ potential at every point (x, y): this function is, by definition, the potential of the vector field F (x, y). Example - Find the potential of the conservative vector field F (x, y) = yi + xj. We choose the origin as the reference point and we set the potential to be equal to an arbitrary constant C there: φ(0, 0) = C. To evaluate the potential at a generic point P1 = (x1 , y1 ) we choose as path a straight line r(t) from the origin to the point P1 parametrised by 0 ≤ t ≤ 1. r(t) = tx1 i + ty1 j, The potential is given by Z Z 1 dr F [r(t)] · dt + C = φ(x1 , y1 ) = F · ds + C = dt r 0 Z 1 (ty1 i + tx1 j) · (x1 i + y1 j) dt + C = x1 y1 + C. 0 3.3.2 Differential method The potential φ(x, y) of the conservative field F (x, y) = F1 (x, y)i + F2 (x, y)j must satisfy the two differential equations ∂φ = −F1 (x, y), ∂x ∂φ = −F2 (x, y). ∂y (7) (8) MA156 - Conservative vector fields 10 We can integrate (7) with respect to x, considering y as a parameter. Its solution is Z φ(x, y) = − F1 (x, y) dx + ψ(y), (9) where ψ(y) is the integration “constant”: notice that it is constant with respect to x, but may still be a function of the other variable y. We can now substitute (9) into (8) to obtain an ordinary differential equation for ψ(y): Z ∂ dψ F1 (x, y) dx. = −F2 (x, y) + dy ∂y Example - Find the potential of the conservative field F = yi + xj. We are looking for a function φ(x, y) such that ∂φ = −y, ∂x ∂φ = −x. ∂y The solution of the first equation, obtained by integrating over x while considering y as a constant is φ(x, y) = −xy + ψ(y). If we substitute this expression for φ into the second equation we obtain an ordinary differential equation for ψ(y): dψ dψ ∂φ = −x + = −x =⇒ = 0 =⇒ ψ(y) = C, ∂y dy dy where C is an arbitrary real constant. Therefore the potential of the conservative vector field F = yi + xj is φ(x, y) = −xy + C, the same result obtained using the previous method. 4 Stokes theorem 4.1 Introduction In this section we will discuss how to find a simple condition to determine whether or not a given vector field is conservative. For example, we know that if a two dimensional vector field, F (x, y) = F1 (x, y)i + F2 (x, y)j, is conservative then ∂x F2 = ∂y F1 . (10) However, we also know that this condition is not sufficient, per se. A field can satisfy it and yet not be conservative. Is it possible to generalise Equation (10), so that it is not only necessary, but also sufficient? The answer is yes and Stokes theorem tells us what we need to do. However, before being able to answer this question we need to introduce a first “derivative” of a vector field, the curl of a vector field. MA156 - Conservative vector fields 11 4.2 The curl of a vector field All the “derivatives” of vector fields are based on the operator ∇, nabla or grad. This symbol can be considered as a vector operator ∇= ∂ ∂ ∂ i+ j+ k, ∂x ∂y ∂z that acts on functions of (x, y, z). The gradient of a scalar function, for example, is the result of applying the operator grad to the function itself: ∂ ∂ ∂f ∂f ∂f ∂ i+ j+ k f (x, y, z) = i+ j+ k. ∇f = ∂x ∂y ∂z ∂x ∂y ∂z This same approach can be used to introduce new “derivatives”, that instead of differentiating scalar functions, like f (x, y, z), differentiate vector fields. One of these derivatives is the curl of a vector field, F (x, y, z). Definition - The curl of a differentiable vector field F is the vector field ∇ × F . Equivalent ways of evaluating this “derivative” are: ∂ ∂ ∂ i+ j+ k × (F1 i + F2 j + F3 k) ∇×F = ∂x ∂y ∂z i j k = ∂x ∂y ∂z F1 F2 F3 ∂F1 ∂F3 ∂F2 ∂F1 ∂F3 ∂F2 − +j − +k − = i ∂y ∂z ∂z ∂x ∂x ∂y The curl of a vector field is related to rotation, it “measures” how the field rotates, swirls at different points of space. Consider, for example, the rotation of a rigid body with constant angular velocity, ω = ω0 k. The velocity at a point (x, y, z) is given by v(x, y, z) = ω × r = −yω0 i + xω0 j. The curl of this vector field is i j k ∂y ∂z = kω0 [∂x x − ∂y (−y)] = 2ω0 k. ∇ × v = ∂x −yω0 xω0 0 The curl of the velocity field is twice the angular velocity of the rigid body. MA156 - Conservative vector fields 12 y x F=0 Figure 5: A centrally symmetric field has zero curl. Properties of the curl 1. ∇ × (gF ) = g(∇ × F ) = ∇g × F , where g(x, y, z) is a scalar function and F (x, y, z) a vector field. 2. The curl of a conservative vector field is zero: ∇ × (∇φ) = 0. This property is the three dimensional version of equation (10). It says that if F is a conservative vector field, i.e. F = ∇φ, then its curl is zero: F conservative =⇒ ∇ × F = 0. Stokes theorem allows us to invert the direction of the arrow, provided that F satisfies some other properties. 3. The curl of a centrally symmetric field is zero: F = f (r)r =⇒ ∇ × F = 0. This results is in line with the interpretation of the curl as a measure of the rotation of a vector field. A centrally symmetric field does not “rotate” and its curl is zero (see Figure 5). 4.3 Stokes theorem We have defined a derivative of a vector field, the curl. Before stating Stokes theorem we need to specify the surfaces we wish to work with. The first requirement is that the surface is orientable, so that we can evaluate flux integrals across it. The second requirement is that the surface must have a boundary, i.e. it must be delimited by a piecewise smooth curve, eventually composed of many bits and pieces (see Figure 6). Finally we need to define a positive orientation for the boundary and the vector normal to the surface. The orientation of the boundary is positive if walking along it the surface is on our left. The MA156 - Conservative vector fields 13 z z z n y n y y x x x Figure 6: A sphere is a surface with no boundary: it is not delimited by a curve. The other two surfaces have boundaries formed by one piece (centre) or two (right). The inner boundary is traversed in the opposite direction of the outer boundary. vector normal to the surface is defined to point upwards. This rules implies that the borders of holes in the surface are traversed in opposite direction to the outer rim, Figure 6, or that the bases of a cylinder are traversed in opposite ways, Figure 7. We are now ready to state Stokes theorem and its two dimension version, Green’s theorem. Theorem (Stokes) Let S be a piecewise smooth, oriented surface in three dimensions, having unit normal n̂ and boundary c consisting of one or more piecewise smooth, closed curves with positive orientation. Then Z Z F · ds = ∇ × F · dS c S Theorem (Green) Let R be a closed region in the xy-plane whose boundary c consists of one or more piecewise smooth non self-intersecting closed curves that are positively oriented. If F = F1 i + F2 j is a differentiable vector field on R then Z Z ∂F2 ∂F1 − dA. F · ds = ∂x ∂y c R R Example - Evaluate S ∇ × F · dS over the hemisphere x2 + y 2 + z 2 = a2 , z ≥ 0, where F (r) = yi + xzj + yk. The boundary of the surface is the circumference x2 + y 2 = a2 , z = 0, (see Figure 8) which can be parametrised by the curve r(t) = a cos(t)i + a sin(t)j 0 ≤ t ≤ 2π . Stokes’ theorem states that Z Z 2π Z dr = ∇ × F · dS = F · ds = dt F [r(t)] · dt S c 0 Z 2π dt [a sin(t)i + a sin(t)k] · [−a sin(t)i + a cos(t)j]dt = −πa2 0 Remark - A consequence of Stokes’ theorem is that the flux of the curl across all the surfaces that have the curve c(t) as border is the same. Therefore, instead of evaluating the flux of ∇ × F over the hemisphere S, we can evaluate the flux across the base B (a much simpler integral) and obtain the same result: Z Z ∇ × F · dS = ∇ × F · dS = −πa2 S B MA156 - Conservative vector fields 14 z S n y r B x Figure 7: The bases of a cylinder are traversed Figure 8: The flux of ∇ × F across S is the same in opposite directions. The vector normal to the surface points outwards. as that across B and as the line integral along r. 4.4 Stokes theorem and conservative fields. When studying conservative fields we have established that a field on a connected domain D is conservative if and only if I F · ds = 0 c for all closed paths c in D. This theorem while giving a complete characterisation of a conservative vector field is rather cumbersome to use: there are infinite closed paths. Stokes theorem allows us to get around this problem. If D is a simply connected domain then any closed path c is the border of an orientable surface S. To each of these surfaces is possible to apply Stokes theorem, Z Z F · ds = ∇ × F · dS. c S If ∇ × F = 0 everywhere in a simply connected domain D then Z ∇ × F · dS = 0, for all surfaces =⇒ S Z F · ds = 0, for all closed path =⇒ F is conservative. c On the other hand we have already stated that if F is conservative then its curl is zero. We therefore a simple criterion to establish whether a field is conservative: A vector field F defined on a simply connected domain D is conservative if and only if ∇×F =0 in D. MA156 - Conservative vector fields 15 For example, the Biot-Savart field, F = −yi + xj , x2 + y 2 (x, y) 6= (0, 0), is not conservative even though ∇ × F = 0, because its domain of definition, the xy-plane minus the origin, is not simply connected. 5 Divergence and Divergence theorem There are two types of derivatives of a vector field. We have seen the curl, ∇ × F obtained by taking the cross product of the grad operator, ∇ with a vector field F . By taking the dot product instead, we obtain a second type of derivative, the divergence: ∇ · F . Definition - The Divergence of a differentiable vector field F (x, y, z) is the scalar function ∇ · F : ∂ ∂ ∂ i+ j+ k · (F1 i + F2 j + F3 k) ∇·F = = ∂x ∂y ∂z ∂F1 ∂F2 ∂F3 + + . = ∂x ∂y ∂x While the curl ∇ × F is a vector, the divergence is a scalar function. The key to the physical meaning of the divergence of a vector field is given by the divergence theorem. The divergence theorem relates surface integrals of vector fields to volume integrals of their divergence. It allows • To get a clearer idea of the physical meaning of divergence. • To express physical laws as relations between derivatives of vector fields. • To find the value of a surface integral by evaluating a volume integral (the latter is usually simpler than the former). Theorem - Let F (r) be a a vector field defined in a volume V . Let S be the orientable surface that encloses the volume V . If F (r) is differentiable in V then Z Z ∇ · F dV = F · dS . V S In words, the volume integral of the divergence is equal to the flux of the vector field. Example - Evaluate the flux of the vector field F (x, y, z) = yi across the sphere of radius one centred at the origin and compare the result with the volume integral of the divergence of F . The sphere is represented by the vector function r(φ, θ) = cos(φ) sin(θ)i + sin(φ) sin(θ)j + cos(θ)k, 0 ≤ φ ≤ 2π, 0 ≤ θ ≤ π. The surface element is dS = [cos(φ) sin(θ)i + sin(φ) sin(θ)j + cos(θ)k] sin(θ)dθdφ. MA156 - Conservative vector fields 16 The flux of F = yi is Z Z F · dS = F [r(φ, θ)] · dS = S S Z 2π Z π dθ dφ [sin(φ) sin(θ)i] · [cos(φ) sin(θ)i + sin(φ) sin(θ)j + cos(θ)k] sin(θ) = 0 0 Z 2π Z π dθ dφ cos(φ) sin(φ) sin3 (θ) = 0. 0 0 On the other hand the divergence of F = F1 i + F2 j + F3 k is ∂F1 ∂F2 ∂F3 ∂y + + = = 0, ∂x ∂y ∂z ∂x ∇·F = so that the volume integral of the divergence is Z Z ∇ · F dV = 0 dV = 0 V V as stated by the divergence theorem. 6 Physical Applications of the divergence theorem 6.1 Divergence and the sources of vector fields The divergence of a field is associated with the sources and the sinks of the field (see Figure 9). A source is a region in space from which field lines flow outward (for example, the neighbourhood of a positive charge or a “source” of water). A sink is a region of space where the field lines converge to (for example, the neighbourhood of a negative charge or a “hole” where water disappears). Consider a small region in a vector field . The volume integral of the divergence is approximately: Z ∇ · F dV ' ∇ · F ∆V , V where ∆V is the volume of the small region. The divergence theorem states that the value of the divergence is related to the flux across the surface of the region: Z Z Z ∇ · F dV = F · dS =⇒ ∇ · F ∆V ' F · dS . V S S The field flows outward form a source, its flux is positive and so is its divergence. The field flows inward towards a sink, its flux is negative and so is its divergence. A positive value of the divergence is associated to the sources of the field, a negative one to its sinks. In other words, the divergence of a vector field is a measure of the inflow or outflow of the field from a small region of space. 6.2 Divergence and Conservation laws The divergence of a vector field is an extremely useful tool to express conservation laws. The total mass of fluid contained in a volume V can change only if there is a mass outflow or inflow across the MA156 - Conservative vector fields 17 Source Sink Positive Divergence Negative Divergence Figure 9: Field lines and equipotential for two equal and opposite point charges. The divergence is negative around the negative charge (sink), positive around the positive one (source). surface S that delimits the volume. This statement can be expressed in integral form as: d (Mass inside V ) = −(Mass outflowing through S) =⇒ dt Z Z ∂ρ + (ρv) · dS = 0 , V ∂t S where ρ(r, t) is the mass density of the fluid and v(r, t) is the speed of a fluid particle at the point r = xi + yj + zk. This relation involves integrals. It can be put in a differential form (an equation that contains only derivatives of the field), by making use of the divergence theorem: Z Z (ρv) · dS = ∇ · (ρv)dV . S V The mass conservation law becomes: Z V ∂ρ + ∇ · (ρv) dV = 0 ∂t This relation must hold for all volumes V . This is possible only if the integrand itself is zero: ∂ρ + ∇ · (ρv) = 0 . ∂t This equation is the mass conservation law expressed in differential form. It shows that the divergence of a field is usually associated with a conserved quantity. For example, an equation exactly analogous to this expresses the charge conservation law in electromagnetism. MA156 - Conservative vector fields 18 6.3 Divergence and electrostatic Gauss law states that the flux of the electric field across a closed surface S is equal to the charge contained in the volume V enclosed by the surface: Z Z 1 ρ(r)dV = E · dS , 0 V S where ρ(r) is the charge density and E(r) is the electric field. This equation can be put in differential form by applying the same procedure as in the previous example: Z Z Z ρ(r) E · dS = ∇ · EdV =⇒ − ∇ · E dV = 0 0 S V V This relation must hold for all volumes. This is possible only if the integrand is zero: ∇·E = ρ 0 This is the first of Maxwell’s equations. This equation can be written in a simpler form: since the electrostatic field E is conservative, E = −∇φ, we can rewrite Maxwell’s equation as an equation for the potential, φ. The advantage of doing so is that only scalar functions appear in it, instead of vectors. ∇ · (−∇φ) = ρ 0 ⇒ ∇2 φ = − ρ 0 Poisson’s equation where the symbol ∇2 φ (called Laplacian) means ∇2 φ = ∂2φ ∂2φ ∂2φ + 2 + 2. ∂x2 ∂y ∂z The equation for the potential in absence of charges, ρ = 0, is ∇2 φ = 0 Laplace equation These last two equations appear in many branches of physics, from the theory of elasticity, to electromagnetism, to quantum mechanics.