Lecture 1b. Differential operators and orthogonal coordinates. Partial derivatives. Divergence and divergence theorem. Gradient. A y. + A y y dy. 1b.

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1 b. Partial erivatives Lecture b Differential operators an orthogonal coorinates Recall from our calculus courses that the erivative of a function can be efine as f ()=lim 0 or using the central ifference form: f ()=lim 0 f (+) f () f (+/) f ( /) The partial erivative of a function of several variables is efine in a similar manner b varing onl one of the variables, such as f (, )=lim 0 f (, +) f (, ) The other variables remain fie. Multiple erivatives can be efine accoringl. For eample, f f (+, ) (, ) f (, )=lim 0 where f / is efine in (3). From the efinition it follows that the orer in which the erivatives are taken oes not matter. For eample () () (3) (4) f = f (5) because in both cases we have the limit as 0 of Graient f (+, +)+ f (, ) f (+, ) f (, +) (6) Assume a scalar fiel f (,, ) is efine in some region of space. If we start at the point (,, ) an move a isplacement l=â +â +â awa, the function value will change from f to f +f. The change is given b f = f + f + f (7) This motivates us to efine a vector fiel, whice call the graient of f, b the formula f f =^a +^a f +^a f We can abstractl think of the el operator as Provie f satisfies some basic continuit/ifferentiabilit conitions. (8) =^a +^a +^a The el operator applie to a scalar fiel prouces the graient. For an isplacement l we can write (9) f = f l (0) For a fie length isplacement (l) this ot prouct will be greatest if l is parallel to the graient. Therefore the irection of the graient is the irection of the maimum rate of change of the scalar fiel with respect to isplacement, an the magnitue of the graient is the corresponing rate of change. This gives us a phsical interpretation of the graient that is inepenent of the coorinate sstem. Divergence an ivergence theorem The ot prouct of the el operator with a vector fiel gives a scalar fiel calle the ivergence of the vector fiel. We have A= A + A + A () We enote the flu of A through the close surface S b ψ A = S A s () The ivergence theorem states that the surface flu is the volume integral of the ivergence V A= S A s (3) If our volume is small enough that A is effectivel constant, then the volume integral reuces to V A an A V A s (4) S We have the phsical interpretation that ivergence is surface flu per unit volume. Note that this interpretation is inepenent of an particular coorinate sstem. The mathematical epression, however, will epen on the A A + A Figure : Divergence for a small cubic surface.

2 b. coorinate sstem. Let's consier the ivergence theorem in rectangular coorinates applie to a bo of imensions,,. If A is the component of A "entering" the, face at =0, then we have an inwar flu of A. At = the fiel is A +( A / ) so we have an outwar flu of [ A +( A / )]. The net outwar flu is ( A / ) =( A / ). Repeating in the an irections we obtain for the flu. A = ( A + A + A (5) ) Curl an Stoke's theorem The cross prouct of the el operator with a vector fiel gives a vector calle the curl of the vector fiel. A= ^a ( A A ) + ^a ( A (6) + ^a ( A A ) A ) Stoke's theorem states that the integral of a vector fiel aroun a close loop equals the integral of the curl over an surface boune b that loop, or S ( A) s= L A l (7) For a ver small, flat surface of area S over which A can be consiere constant, we have S ( A) s ( A) S. This is illustrate below. If we call L A l the rotation of A about the curve L then the phsical significance of the curl is that A is the maimum rotation per unit area, an the rotation has this value in the plane normal to A. A giving A=â A, but if ( A A 0 then A ) will have a component. The conition A A=0 hols everwhere onl if the irection of A is the same everwhere. In the above illustration, we will get the most rotation about the curve if we orient S to be parallel to A. For this orientation we have A S A l. The magnitue of L the curl is the maimum rotation per unit area, an the rotation is maimum in a plane normal to the irection of A. This phsical interpretation is inepenent of an particular coorinate sstem. As alwas, the mathematical epression of this interpretation will epen on the coorinate sstem emploe. A possible misconception of the curl of a vector fiel is that A 0 implies that the irection of the fiel A somehow "curls aroun" the vector A. This is not necessaril the case. For eample, A=â gives A=â ( ), but the irection of A oes not make circles aroun the ais. Instea, if we integrate A l aroun a circle we get a non-ero value. Figure 3: This vector fiel v has curl â ( ). The erivation of the component of the curl is shown below. v=â A + A S l A A + A A A Figure : Curl of a vector fiel. Notice that if A=â A everwhere then â A=0. In general, if the irection of A is constant then A has no component parallel to A an A A=0. Keep in min that A A=0 is not an ientit. For eample, at a single point in space we might have A =A =0 Figure 4: A l for a small rectangle in the - plane. If we integrate A l counter-clockwise aroun the rectangle of imensions, we get a positive contribution A on the bottom sie an a negative contribution (A + A / ) on the top sie. The net is A /. Likewise, we get a positive contribution

3 b.3 (A + A / ) from the right sie an a negative contribution A. The net is A /. The total line integral is ( A / A / ). This is ( A) â. Laplacian The ivergence of the graient of a scalar function is calle the Laplacian of the function an is enote b f. We have f = ( f )= f + f + f (8) In rectangular coorinates the Laplacian is the sum of secon erivatives. The secon erivative is the erivative of the erivative, an using the central-ifference efinition of a erivative we have (vali as 0 ) f ()= f f () (+/) f ( /) Using the central-ifference efinition again gives us () ( f (+) f ( ) f () f ( ) f ) ( ) = [ f (+)+ f ( )] { f ()} (9) (0) We see that the secon erivative is relate to the ifference between the function value at a point an the average value nearb. Doing the same for the an coorinates we obtain f 6 ( f (+,, )+ f (,, ) [ + f (, +, )+ f (,, ) f (,, 6 + f (,, +)+ f (,, )) )] () Therefore, the Laplacian of a function measures the ifference between the function at a point an the function's average value at neighboring points. More precisel, the Laplacian is 6/ times the ifference between the average value of the function on a sphere of (ver small) raius an the value of the function at the sphere center. Inee, if f =0 everwhere, then the average value of f over a sphere (of an sie) is equal to the value of f at the sphere's center (this is calle "Gauss's harmonic function theorem"). In rectangular coorinates, the Laplacian of a vector is efine as a vector whose components are the Laplacians of the corresponing components of the vector. A â A +â A +â A () Note that this efinition is specific to rectangular coorinates. In other coorinate sstems the Laplacian of a vector is not so simple. We will nee to revisit this when we consier spherical coorinates. General orthogonal coorinates Consier an orthogonal coorinate sstem where spatial position r is etermine b the three coorinates u, v, w. We tpicall specif three functions (u,v, w), (u, v, w) an ( u, v, w) that give the rectangular coorinates as functions of u,v, w. As an eample =usin (v)cos(w) =usin (v)sin(w) =u cos(v) (3) efines the spherical coorinates u, v, w (whice usuall enote as r, θ, ϕ ). B orthogonal we mean that the unit vectors â u = r/ u r/ u â v = r/ v r/ v â w = r/ w r/ w (4) are orthogonal at all points in space, that is, ^a u ^a v =^a u ^a w = ^a v ^a w =0 everwhere. Let's efine = r (5) u an likewise for,. We call these the metric coefficients. The phsical significance of is that the length of the isplacement r ue to a change in the coorinate u of magnitue u is l = u. Let the coorinates change b the ifferential amounts u,,w. The length of the resulting isplacement, call it l, will be given b the Pthagorean formula l = u + + w (6) For eample, in rectangular coorinates l = + + (7) an we see that h =h =h =. In clinrical coorinates l =() ρ +(ρ) ϕ +() (8) so h ρ =h =,h ϕ =ρ. In spherical coorinates l =() r +(r) θ +(r sin θ) ϕ (9) so h r =, h θ =r,h ϕ =r sin θ. The metric coefficients etermine the specific forms that the ifferential operators take, as we will now see. Graient The graient is a irectional erivative. The value of ( f ) â is equal to f /l in the irection of â. In the The Pthagorean theorem applies onl if the coorinates are orthogonal. Non-orthogonal coorinates woul have cross-terms such as v u.

4 b.4 irection â u, l= u, so the u component of f is ( f / u)/ an similarl for the v an w components. Therefore f f =^a u u +^a f v v + ^a f w w (30) This epresses the graient in an orthogonal coorinate sstem. Divergence The phsical significance of the ivergence is that A is the net outwar flu per unit volume of the vector fiel A from an infinitesimal volume V. Consier a volume prouce b the coorinate changes u,,w. The lengths of the sies will be u,, w so the volume will be V = u w (note the istinction between V an ). Consier the flu in the u irection. The flu into the volume will be w evaluate at u. The flu out of the volume will be w evaluate at u+u. The net flu in the â u irection is We can write this as [( ) u+u ( ) u] w = u ( )u w (3) Laplacian Since the Laplacian is the ivergence of the graient, we can etermine its epression using our two previous results. Substituting the u,v,w components of (30) for, A v, A w in (33) we obtain f = u( f u ) Curl v( f v ) w( h v f w) (34) The curl is the rotation per unit area. Let's consier the component of rotation about the w ais. As illustrate below, consier the contribution of A v to this. A v u u+u u u+u w u ( )V (3) Appling the same iea to the remaining coorinates we have A= u ( ) v ( A v ) w ( A w ) + u u Figure 5: Calculating the u contribution to ivergence. (33) as our epression for the ivergence in general orthogonal coorinates. Figure 6: Calculating the A v contribution to the w component of curl. For rotation in the irection shown, at u+ u we have a contribution to A l of (A v ) u+u. At u the contribution is (A v ) u. The total contribution ue to A v is (A v ) u+u (A v ) u = u (h A v v )u. contributes ( u) v ( u) v+ = v (h u )u The area is u so the w component of A is [ u ( A v ) v ( )] (35) Doing the same for the u an v components we obtain A= ^a u [ v ( A w) + ^a v [ w ( ) + ^a w [ u ( A v ) w ( A v )] u ( A w )] v ( )] (36)

5 b.5 Useful ifferential operator ientities The following ientities can be irectl verifie. The ivergence of the curl of a vector fiel is ienticall ero ( A )=0 (37) The curl of the graient of a scalar fiel is ienticall ero. f =0 (38) Recall the prouct rule for scalar erivatives: ( fg )'= f g ' + f ' g. Here are prouct rules involving graient, ivergence an curl. ( fg )= f g+( f )g (39) ( f A)= f A+ f A (40) ( f A)= f A+ f A (4) The following formula for the ivergence of a cross prouct will be important for us A B =B A A B (4) We will make etensive use of the following ientit A= ( A) A (43) This can be irectl verifie in rectangular coorinates. We can rewrite this as A ( A) A (44) This will serve as the efinition of the Laplacian of a vector fiel in an coorinate sstem. In rectangular coorinates it works out to be (), but for a general sstem of orthogonal coorinates u, v, w it oes not work out so cleanl. In general A â u +â v A v +â w A w (45) In non-rectangular sstems we must use (44) to escribe the Laplacian of a vector. References. Kresig, Aance Engineering Mathematics, 4 th E.,Wile, 979, ISBN Hobson, E. W., The Theor of Spherical an Ellipsoial Harmonics, Chelsea, Buak, B. M., A. A. Samarskii an A. N. Tikhonov, Collection of Problems in Mathematical Phsics, Dover, 988, ISBN (See Supplements.) 4. Lebeev, N. N., I. P. Skalskaa an Y. S. Uflan, Worke Problems in Applie Mathematics, Dover, 979, ISBN (See Chapter 7 Curvilinear Coorinates.) 5. mathworl.wolfram.com/orthogonalcoorinatesstem.html Appeni Orthogonal coorinate sstems The Helmholt equation is known to be separable in eight coorinate sstems in aition to rectangular, clinrical an spherical coorinates [4]. Some of these are Elliptic clinrical coorinates =acoscos v =a sinsin v =w Here a is a fie parameter chosen to give a esire elliptical geometr an u, v, w are the coorinates. These are similar to (circular) clinrical coorinates ecept that raial istance r has been replace b a cos for the coorinate an a sin for the coorinate. For fie u, ( a cos) +( =cos asin) v+sin v= is the equation of an ellipse with semi-major/minor aes a cos an a sin. For fie v, ( a cos v) ( =cosh a sinv) u sinh u= is the equation of a hperbola. Bipolar clinrical coorinates = a sin cos cosv asin v = cos cos v =w Again, a is a fie parameter. A surface of constant u is a circular cliner of raius r= a/ sin an center =acos/ sin, =0. These coorinates are useful for escribing a two-wire transmission line. Spheroial coorinates There are two spheroial coorinate sstem relate to spherical coorinates b stretching or compressing the coorinate. The prolate spheroial coorinates are efine b =asin sin v cos w =a sinsin v sin w =a cos cosv these are similar to the spherical coorinates (with v=θ, w=ϕ ) but with r=a sin for the, coorinates an r=a cos for the coorinates. Since cos>sin the coorinate will be larger (relative to the, coorinates) than it woul be in spherical coorinates. The result is that a surface of constant u is a spheroi that is stretche in the irection. On the other han, for the oblate spheroial coorinates =acossin v cos w =a cossin vsin w =asincos v the situation is reverse; the coorinate is relativit smaller.