Lecture 12 Vector Analysis I (See Chapter 6 in Boas)
Now we want to combine our earlier discussion of vectors with our more recent discussion of partial derivatives and functions of several variables. Recall from Lecture 6 that we have defined various products for vectors. The first is the scalar product,
3 rrxx12 12 yy 12 zz 12 rr 1,2,kk rr 12cos 12 . (12.1) k1
An example of how this is applied in physics is when calculating the work done by a force F on an object that moves through a path element ds , dW F ds .
The second product is the vector or cross product that produces a vector orthogonal to both of the original vectors with a sense provided by the right-hand-rule,
rr12 yz 12 yzx 21 ˆˆ zx 1221 zxy xy 12 xyz 21 ˆ, 3 (12.2) r r r r r r . 1 2k klm 1, l 2, m klm 1, l 2, m lm,1
An example of how this quantity arises in physics is in the analysis of motion in a circle. Consider a particle with position defined by r moving in a circle of radius rsin at constant angular velocity with the origin of coordinates on the perpendicular through the center of the circle as indicated in the figure. We associate this trajectory with a vector valued angular velocity with magnitude and vector direction orthogonal to the plane of the circle with sense give by the right-hand-rule applied to the motion about the circle. Then we have the usual linear vector velocity given by v r r , which is tangent at each point to the circle describing the path of motion.
A third product arises from combining the first two. This yields the triple scalar product defined by
3 rrr1 2 3 klm rrr 1, k 2, l 3, m rrrrrr 3 1 2 2 3 1 . (12.3) k, l , m 1
Physics 227 Lecture 12 1 Autumn 2008 This product will arise, for example, when we take the component of a torque, rF, along a vector n , n n r F .
Something we did not mention in Lecture 6 is the triple vector product defined by
rrr123 rrrrrr 231132123 rrr . (12.4)
Remarkably, we can obtain an understanding of the right-hand-side of this last equation by (effectively) pure thought. We apply a very useful technique that can be called either the “What else can it be?” theorem or the “Only game in town” theorem. The basic idea is that we want to define a vector constructed out of the 3 vectors rr, 12 and r3 , which has the following general properties: a) the form of the vector can only know about the vector directions defined by and , and cannot depend on any specific choice of basis vectors; b) due to the structure of the left-hand-side of Eq. (12.4), each term on the right-hand-side must be linear in each of the vectors; c) as a vector the right-hand-side must be orthogonal to r1 (due to the left-most or last cross product). Thus pure thought says the most general allowed form is
r1 r 2 r 3 c r 1 r 3 r 2 r 1 r 2 r 3, (12.5) where the constant c is independent of the specific vectors. Each term has the required properties and we have arranged that the right-hand-side is orthogonal to r1 , leading to the minus sign. We can determine c by considering a specific example, say r1 xˆ,, r 2 x ˆ r 3 y ˆ . Explicit calculation yields r1 r 2 r 3 xˆˆ zˆ y and c r1 r 3 r 2 r 1 r 2 r 3 c yˆ so that c 1.
Such a product will arise in the case of circular motion when we evaluate the angular momentum, L r p r mv mr r .
Since things get interesting when we consider variation with time, we need to develop the required notation for time derivatives of vectors (something we have already used). If we use a fixed set of basis vectors, i.e., the basis vectors are not themselves functions of time ( xˆˆ y zˆ 0), we have
Physics 227 Lecture 12 2 Autumn 2008 r xxˆˆ yy zzˆ, dr dx dy dz v r xxˆ yy ˆ zzˆ xx ˆ yy ˆ zz ˆ x ˆ y ˆ z ˆ, (12.6) dt dt dt dt d2 r d 2 x d 2 y d 2 z a r xxˆ yy ˆ zzˆˆ x ˆ y ˆ z. dt2 dt 2 dt 2 dt 2
We also have various versions of the chain rule. For a product of a scalar (function of time) times a vector (function of time) we have
d dA da aA a A. (12.7) dt dt dt
Similarly we have the two forms
d dA dB ABBA , dt dt dt (12.8) d dA dB ABBA , dt dt dt where in the first line the order of the factors in each term does not matter, while in the second line the order matters (i.e., there is a potential factor of -1 in the cross product).
Once more we will illustrate some of these ideas by considering uniform motion in a circle. With the geometry as defined in the earlier figure moving on a circle means 2 2 2 2 r r r r0 0sin 0 , where r0,, 0 0 are constants. Uniform motion means 2 2 2 vr 0 , with 0 a (scalar) constant. Taking time derivatives of these quantities we obtain the following properties of the various vectors
dd22 r r0 0 2 r r 2 v r v r , dt dt dd22 v v0 0 2 v v 2 a v a v , (12.9) dt dt a v r r r r.
Physics 227 Lecture 12 3 Autumn 2008 In words, we say that for uniform circular motion the acceleration is orthogonal to the velocity which is itself orthogonal to the position vector. Taking one more derivative we have
dv 2 v r 0. v v a r a rˆ (12.10) dt r
In the special case that we take the origin to be at the center of the circular orbit so that r0 is the radius of the circle, we have the familiar result that the radial (or 2 centripetal) acceleration is given by ar v r0 .
Circular motion is a natural setting for the use of curvilinear coordinates, but it is important to recall that for such coordinates the unit basis vectors themselves are functions of the coordinates (and will likely be time dependent). The most useful curvilinear basis vectors are those for cylindrical coordinates and for spherical coordinates. Compared to the rectilinear basis vectors, the cylindrical basis vectors look like
zz xy22 0 x cosy , tan1 0 2 y sin x xxˆˆ yy Basis Vectors: zˆ ,ˆ xˆˆ cos y sin , ˆ xyˆˆsin cos , (12.11) zzˆˆˆˆ ˆˆ 0, ˆ ˆ zˆ plus cyclical permutations Path Length: ds dx xˆˆ dy y dz zˆ ds d ˆ d ˆ dz zˆ, ds2 dx 2 dy 2 dz 2 ds 2 d 2 2 d 2 dz 2.
The corresponding spherical basis vectors are
Physics 227 Lecture 12 4 Autumn 2008 r x2 y 2 z 2 0 r zr cos xy22 xrsin cos tan1 0 , z yr sin sin y tan1 0 2 x xxˆˆ yy zzˆ Basis Vectors: rˆ , r sin cos xˆˆ sin sin y cos zˆ sinˆ cos zˆ , ˆ , cos cos xˆˆ cos sin y sin zˆ cosˆ sin, zˆ ˆ xyˆˆsin cos , rrˆˆˆ ˆ ˆ ˆ 0 ˆˆrˆ plus cyclical permutations Path Length: (12.12) ds dx xˆ dy y ˆ dz zˆ ds dr r ˆ rd ˆˆ rsin d , ds2222 dx dy dz ds 2222222 dr r d rsin d .
Note that the “azimuthal” angle is the same periodic angle in both cylindrical and spherical coordinate systems ( 02, with the endpoints of the interval identified), but that the spherical polar angle varies only on the limited range 0 to ( 0 ). All of the various unit vector sets form right-handed, orthogonal triplets.
ASIDE: For future reference we will also define the case for general 3D orthogonal, right-handed, curvilinear coordinates, unit vectors and pre-factors (the hk) defined by the path length,
xxx123,,,,. eeeˆ 123 ˆ ˆ ds hdxe 111 ˆ hdxe 222 ˆ hdxe 333 ˆ (12.13)
For example, for cylindrical coordinates we have x1,, x 2 x 3 z ,
Physics 227 Lecture 12 5 Autumn 2008 h1 h 3 1, h 2 , and for spherical coordinates we have x1 r,, x 2 x 3 ,
h11, h 2 r , h 3 r sin .
As a first illustration we return to the familiar case of motion on a circle, again with the geometry of the previous figure. In cylindrical coordinates we place the circle in the x-y plane at fixed zz 0 , or in spherical coordinates at fixed 0 . (The simplest case is z0 0 , 0 2 .) For uniform motion we define tt 0 and identify
rˆ 0, t sin 0 ˆ t cos 0 zˆ
sin00 cos xˆˆ sin y cos zˆ , dd rˆ 0, t sin 0 ˆ t sin 0 sin x ˆ cos y ˆ dt dt ˆ sin 0 t , (12.14) ˆ 0, t cos 0 ˆ t sin 0 zˆ , ddˆˆ cos 00 ˆ cos tz ˆ 0 , dt dt d ˆ cos xyˆˆ sin ˆ. dt
Finally, as earlier, we set the radius of the circle to a constant,
r 0 r 0 rˆ 0 r 0sin 0 ˆˆ cos 0 r 0 zˆˆ 0 z 0 z , (12.15) and find that the motion is described by the time dependence of the ˆ unit basis vector (i.e., rz0,,, 0 0 0 and zˆ are all constant in time)
ˆˆ v r 0 0 ˆ 0 v , ˆ 2 a v r 0 0 ˆˆ 0 0 (12.16) v2 ˆ. 0
Physics 227 Lecture 12 6 Autumn 2008 This last result is just the familiar centripetal acceleration (directed radially inward towards the axis of the circular motion). Note that, as advertised in Eq. (12.9), the ˆ ˆ ˆ location vector r r sin00 cos zˆ is orthogonal to the velocity vector r which is orthogonal to the acceleration vector r ˆ (but r is not necessarily orthogonal to r ).
Now we want to combine the earlier discussions and introduce a notation for partial derivatives that also carries information about the vector direction of the individual partial derivatives. The quantity of interest for this purpose is the (vector-valued) gradient operator. In rectilinear notation we have
xˆˆ y zˆ . x y z (12.17)
If we apply this operator to a scalar-valued function f x,, y z, we obtain a vector f ,
f f f f x,,, y z xˆˆ y zˆ (12.18) x y z whose direction tells us the direction in which increases most rapidly. For any direction defined by a unit vector nˆ , the rate at which varies in that direction is given by the scalar product nfˆ . If we consider a surface defined by f x,, y z f0 , the vector is orthogonal to that surface. In particular, if t is a vector locally tangent to such a surface, then tf 0. For the special case of a surface defined by a linear equation (function), e.g., ax by cz d , we know that the surface is flat (a plane) and the normal given by the gradient ax by cz axˆˆ by czˆ is a constant vector. As expected the vector normal to a plane (a flat 2-D surface) is in the same direction at every point on the plane. In the more general case, where is not a linear function, the surface defined by is not flat and the vector direction of the gradient will vary as we move around on the surface. However, we can still define a “tangent plane” to the surface at any specific point r0 x 0 xˆˆ y 0 y z 0 zˆ via the usual relation between a normal and a plane
Physics 227 Lecture 12 7 Autumn 2008 (local) tangent plane: r r00 N r r f 0. (12.19) r0
Every point r that satisfies Eq. (12.19) lies on the plane tangent to the surface defined by f x,, y z f0 at the point r0 x 0 xˆˆ y 0 y z 0 zˆ (which is presumed to lie on the original surface, i.e., f x0,, y 0 z 0 f 0 ).
We can use the gradient to define the total differential of a function,
f f f df dr f dx dy dz. (12.20) x y z
Clearly the gradient is a useful operator, which we want to be able to express also in curvilinear coordinates. We have (note especially the extra factors in some denominators arising from the change of variables and necessary to provide an interpretation in terms of inverse lengths)
xˆˆ y zˆ x y z 1 ˆ ˆ zˆ z 11 rˆ ˆˆ (12.21) r r r sin eeˆˆeˆ 12 3 . h1 x 1 h 2 x 2 h 3 x 3
With the above definitions we can construct other useful quantities (see Section 6.7 in Boas and your notes on electromagnetism from Phys. 122). If we consider a vector- valued function, e.g., the electric field (Vxyz ,,,, VxVyVzxˆˆ y z ˆ Exyz ), we can define a scalar function from the “divergence”, i.e., the scalar product with the gradient operator (in some sense this quantity is the “outward” flow defined by V ,
Vxz x,,,, y z Vy x,, y z V x y z V x,,. y z (12.22) x y z
Physics 227 Lecture 12 8 Autumn 2008 Because of the coordinate dependent extra factors in the gradient operator for curvilinear coordinates (see Chapter 10 in Boas), care must be taken when evaluating the divergence (and the other quantities below) in curvilinear coordinates. For the familiar case we have
11Vz ,, Vz ,, Vz,, V x,, y z z z 2 1r Vr r,, 1 sinVr , , 1 Vr ,, r2 r rsin r sin (12.23) 1 hhVxxx231123 ,,,, hhVxxx 132123 h h h x x 1 2 3 1 3
h1 h 2 V 3 x 1,, x 2 x 3 . x 3
ASIDE: The extra factors in the general case can be understood as arising from writing the expressions in terms of “divergence-free” basis vectors,
eeˆˆ12 eˆ3 0. (12.24) h2 h 3 h 1 h 3 h 1 h 2
The reader is encouraged to verify these results for cylindrical and spherical coordinates.
Similarly we can define another vector function (field) by using the cross product. This is called the “curl” and, in some sense, defines the circular flow contained in V ,
xˆˆ y zˆ Curl V x , y , z V x , y , z x y z
VVVx y z (12.25) xˆˆ Vz V y y V x V z zˆ V y V x . y z z x x y
Physics 227 Lecture 12 9 Autumn 2008 In cylindrical coordinates the Curl is
Curl V , , z V , , z 11 ˆ (12.26) ˆ Vzz V V V zˆ V V . zz
In the general case we have
h1 eˆ 1 h 2 e ˆ 2 h 3 e ˆ 3 1 Curl V x , y , z V x , y , z , (12.27) h1 h 2 h 3 x 1 x 2 x 3
hV1 1 h 2 V 2 h 3 V 3 which, for the 2 special cases of interest, yields Eq. (12.26) and for spherical cooridnates Curl V r , , V r , , rˆ ˆ 11 sinV V Vr rV rsin r sin r r (12.28) ˆ 1 rV Vr . rr
Next we can create a scalar function from a scalar-valued function with two applications of the gradient operator. This defines the Laplacian operator,
222 2 fff f f x,,. y z (12.29) x2 y 2 z 2
The general form is 2 f f x1,, x 2 x 3 1 h2 h 3 f h 1 h 3 f hh12 f (12.30) . hhhxhx123111 xhx 222 xhx 333
Physics 227 Lecture 12 10 Autumn 2008 So in our 2 special cases the Laplacian is
22 2 11 f f f f f,, z 2 2 2 z
2211 ff f r, , r sin r22 r r r sin (12.31) 1 2 f . r 2sin 2 2
The Laplacian plays a central role in classical physics,
Laplace's Equation: 2 x , y , z 0, 1 2 Wave Equation: 2 x , y , z , t x , y , z , t , ct22 (12.32) 1 Diffusion Equation: 2 x , y , z , t x , y , z , t , 2 t where c is a constant with the units of a velocity (length/time, the speed of the wave) 2 and 2 has the units of length time .
We can use 2 gradients and the triple vector product to define another vector quantity (the curl of the curl)
V x,, y z V 2 V
Vxz x,,,, y z Vy x,, y z V x y z xˆˆ y zˆ x y z x y z (12.33) 2 2 2 ˆˆ 2 2 2 Vx x V y y V z zˆ. x y z
We can also consider other constructions involving 2 gradients. For example, consider the curl of a gradient,
Physics 227 Lecture 12 11 Autumn 2008 xˆˆ y zˆ f x,, y z x y z fff x y z 2f 2 f 2 f 2 f 2 f 2 f xˆˆ y zˆ (12.34) yzzy zxxz xyyx 0.
The final step, i.e., the vanishing of the curl of the gradient, is guaranteed by the fact that the partial derivatives commute, 22f x y f y x, as we discussed in the previous lecture. Note that this vanishing is independent of the specific function f. Any vector-valued function that is obtained by taking the gradient of a scalar function is guaranteed to be curl-free and any curl-free vector function can be expressed as the gradient of a scalar function (in any simply connected region of space where the partial derivatives exist and are continuous, i.e., where the gradient is actually defined – see the next lecture).
As a final 2-gradient quantity consider the (scalar) divergence of a curl,
V x,, y z xˆˆ Vz V y y V x V z zˆ V y V x y z z x x y (12.35) VVVVVVz y x z y x x y z y z x z x y 2VVV 222VV 2 2V x x yy z z 0. y z z y z x x z x y y x
Again we are guaranteed that this quantity vanishes independent of the specific vector field V .
Applying these operators to products of functions requires a careful implementation of the usual chain rule. For example, we have
Physics 227 Lecture 12 12 Autumn 2008
f V f V V f
Vxz x,,,, y z Vy x,, y z V x y z f x y z (12.36) f f f VVV . xx y y z z
In the next lecture we want to turn to the question of “undoing” derivatives in more than one dimension, i.e., performing integrals in multiple dimensions.
Physics 227 Lecture 12 13 Autumn 2008