1 U Deprtment of Physics 301A Mechnics - I. INTEGRAL THEOREM A. Introduction The integrl theorems of mthemticl physics ll hve their origin in the ordinry fundmentl theorem of clculus, i.e. xb x df dx dx f(x b f(x (1 Using this theorem multiple times we my generlize to 2 nd 3 dimensionl geometries (or even beyond!. In ll cses we find the sme generl pttern viz. : The Integrl of derivtive yiel the function itself summed up over ll the boundry points. Algebr in multiple dimensions is fcilitted by using vector nottion such s r(t. o lso, function F(x, y, z my be written F( r. The ordinry rules of prtil differentition govern our clculus e.g. d F dx F ( r(t dt x dt + F dy dt + F dz dt (2 Here lso, judicious choice of nottion helps us condense our expressions. In prticulr, the grdient nottion will be of prticulr use in writing our expressions concisely. ˆx x + ŷ + ẑ (3 note tht the ˆx nd the other corresponding terms represent unit vectors in the corresponding directions. The definition expressed in eqution (3 gives us nottionl tool tht llows us to mrtil our three bsic prtil derivtives efficiently. In prticulr, we my write in hndy compct nottion: df dt F d r dt (4 Assertion: B. The fundmentl theorem for line integrls d r F F ( r f F ( r i (5 Proof: Ask yourself... how would you ctully perform the integrl? The nswer is tht you would supply some prmetriztion of the curve i.e. r(s where r(s i r i nd r(s f r f nd r(s trces out the curve s s [s i, s f ] procee to sweep through its vlues. Then df df ( d r F ( d r F d r F (6 o d r F sf s i df rf r i df F ( r f F ( r i (7 The pttern is universl. In ech cse we consider n integrl nd then sk how we would ctully perform it. A simple prmetriztion le immeditely to the theorem.
2. Green s Theorem in the Plne uppose we hve function Q(x, y nd region in the x y plne bounded by curve. uppose further tht we must evlute Q onsider the bove... how would you ctully do it? In generl, double integrls re evluted s iterted ordinry one-dimensionl integrls. Q Y2(x dx Y 1(x Q dx{q(x, Y 2 (x Q(x, Y 1 (x} dx Q(x, Y 2 (x + dx Q(x, Y 1 (x b From Eqution 8 tht this is just wht we men by the line integrl: Q(x, y dx Implicitly here, we trverse the boundry curve in counter-clockwise (positive mnner unless otherwise noted. In summry then: Q Q dx nd P x P dy Finlly, we note tht by dding these two results we cn configure the resulting identity in very suggestive form: ( Ay x A x (A x dx + A y dy D. tokes Theorem A surfce bounded by curve. onsider How would we do it? A(x, y, z dx A(x, y, z dx A(x, y, z(x, y dx Φ(x, y dx If Φ(x, y A(x, y, z(x, y. Now Φ(x, y dx dxdy Φ
3 nd Φ A + A Notice! The surfce norml to t point (x, y, z is prllel to o (z z(x, y ( x,, 1 ˆn. n y Recll, tht dσ dσ xy o! A dx dxdy dσ ( A + A ( A A ( A n y A n y Now, by dding in the equivlent terms from y nd z components, we chieve: (A x dx + A y dy + A z dz (( dσ n y A x A x (( Az dσ A y n x + ( + ( Ax x A y n x A y n y + A z x ( + A z n y n x ( Ay x A x x A z In modern nottion this ppers substntilly condensed s: A d r A ˆn dσ E. Divergence Theorem For given function R(x, y, z, consider dz R How would you do it? dz R Zh2 (x,y dz R Z h1 (x,y
4 Now we write bove s {R(x, y, z h2 (x, y R(x, y, z h2 (x, y} Now, we hve dσ xy dσ cos(ˆnẑ or dσ dσxy Now, R(x, y, z hh i(x, y is wht we men by upper R nd if we let ˆn lwys men the outwrd pointing unit vector then R(x, y, z b (x, y lower z n R o By ddition ol ol dz R dσ R ( P dz x + Q + R dσ ( P n x + Q n y + R Writing in modern nottion we hve d A A ˆn dσ F. ummry of Integrl Identities At this point we collect our integrl identities for esy reference. 1. The fundmentl theorem of clculus: xb df dx dx f(x b f(x (8 2. The fundmentl theorem for line integrls: x d r F F ( r f F ( r i (9 3. Green s theorem in plne. is plnr re bounded by the curve : ( Ay x A x (A x dx + A y dy (10
5 4. tokes theorem for n rbitrry vector function A( r nd surfce bounded by curve : A ˆn dσ A d r (11 5. The divergence theorem for ny vector function A( r, nd volume bounded by surfce : d A A ˆn dσ (12