Chapter 3 - Vector Calculus Gradient in Cartesian coordinate system f ( x, y, z,...) dr ( dx, dy, dz,...) Then, f f f f,,,... x y z f f f df dx dy dz... f dr x y z df 0 (constant f contour) f dr 0 or f is the direction to this contour, and it is the direction of maximum rate of change in f
Basis vectors in other orthogonal coordinate systems Polar coordinates e er er cose sine 2 e sine cose 2 Spherical coordinates er sin cos e sin sin e cos e 2 3 e cos cos e cos sin e sin e e sine cose 2 3 2 er e e
Differentials of basis vectors in orthogonal coordinate systems Polar coordinates er cosesine2 e er e sine cose 2 r sin cos 2 cos sin 2 de e e d e d de e e d erd d dt d er e, e er dt r rer dr dr er r der er dr e rd For any f(r,) f df f dr dr r f f f er e r r f d Gradient f is the direction of the steepest change in f
Spherical coordinates er e e er sin cos e sin sin e cos e 2 3 e cos cos e cos sin e sin e e sine cose 2 3 2 der e d esin d de er d ecos d de er sin d e cos d r rer dr dr er r der er dr e r d e r sin d d d d er e e sin, e er e cos, e er sin e cos dt dt dt For any f(r,,ϕ) df f f f f dr dr d d r f f f f er e e r r r sin
Scale Factors in three dimensions y dy dx x h, h r r
Scale Factors in three dimensions r dθ h, h r, h r sin r h, h r, h r z
Generally for all orthogonal systems Length element : d r e h dq e h dq e h dq q q2 2 2 q3 3 3 Area element : d s e h dq h dq e h dq h dq e h dq h dq Volume element : dv h dq h dq q 2 2 3 3 q2 3 3 q3 2 2 h dq 2 2 3 3 f f f f( q, q2, q3) eq e q e 2 q3 h q h q h q 2 2 3 3 h i here are the scale factors that turn the increments in the new coordinates into the corresponding lengths. E.g., in spherical coordinates, h h h h r h h r r, 2, 3 sin f f f f er e e r r r sin
Divergence and Curl in Cartesian coordinates Divergence f f2 f3 f ( x, y, z) x y z Example: f ( r) r ( x, y, z) f 3 r Example: f f ( r) ( y, x,0) f 0 dispersion 2 2 2 2 f 2 2 2 x y z f Laplacian f Curl 3 f2 f f3 f2 f f ( x, y, z),, y z z x x y Example: f ( r) ( y, x,0) f (0,0,2) Example: f ( r) r ( x, y, z) f 0 circulation 2 0, f f f f
Generally for all orthogonal systems dr e h dq e h dq e h dq q q2 2 2 q3 3 3 Area element : d s e h dq h dq e h dq h dq e h dq h dq Volume element : dv h dq h dq q 2 2 3 3 q2 3 3 q3 2 2 h dq 2 2 3 3 f f f f ( q, q2, q3) eq e q e 2 q3 h q h q h q 2 3 i, j, k 2 2 3 f fhh 2 3 hh 2h3 q q f eq i h h f h h h q ijk i k k j f h h f h h 2 3 3 2 2 q3 2 h2h3 f h3h f hh 2f f hh 2h3 q h q q2 h2 q2 q3 h3 q3 3
Line integrals Vector field (e.g., force or an EM field) W f dr lim f ( ri) r n i i r n r 0 Similar to a contour integral f () z dz but f ( x, y) is NOT f(x+iy) = f(z) and thus generally no anti-derivative exists. Thus it is generally path-dependent! Fool-proof way to evaluate it is to parametrize the path. ri W is path-independent f ( r) F( r) for some F(r)
Surface integrals I f d s lim f s S Vector field (e.g., E or B) n n i i (flux) ds For an open surface, the direction of ds ( to surface element) must be explicitly defined. For a closed surface, it is outward normal by convention. When the surface is parallel to a plane of Cartesian axes, use dx, dy, dz for the integral. E.g., if ds//e z, For general surface, parametrize it, e.g., by r = r(u,v), and d s S ezdxdy, f d s S f dxdy r r r r d s du dv, f ds f ( r) du dv u v S U u v where U is the domain of (u,v) 3
Surface integral of a scalar field r r I f ( r) ds f ( r) du dv S u v U ds where both the integrand f(r) and the surface element ds are taken to be scalars Example: Surface area of z = g(x,y) r r r( u, v) ( u, v, g( u, v)) (,0, gx), (0,, g y) u v r r ( gx, gy,) u v A ds g g du dv 2 2 x y S T
Divergence and Curl in Cartesian coordinates Divergence f f2 f3 f ( x, y, z) x y z Example: f ( r) r ( x, y, z) f 3 r Example: f f ( r) ( y, x,0) f 0 dispersion 2 2 2 2 f 2 2 2 x y z f Laplacian f Curl 3 f2 f f3 f2 f f ( x, y, z),, y z z x x y Example: f ( r) ( y, x,0) f (0,0,2) Example: f ( r) r ( x, y, z) f 0 circulation 2 0, f f f f
Generally for all orthogonal systems dr e h dq e h dq e h dq q q2 2 2 q3 3 3 Area element : d s e h dq h dq e h dq h dq e h dq h dq Volume element : dv h dq h dq q 2 2 3 3 q2 3 3 q3 2 2 h dq 2 2 3 3 f f f f ( q, q2, q3) eq e q e 2 q3 h q h q h q 2 3 i, j, k 2 2 3 f fhh 2 3 hh 2h3 q q f eq i h h f h h h q ijk i k k j f h h f h h 2 3 3 2 2 q3 2 h2h3 f h3h f hh 2f f hh 2h3 q h q q2 h2 q2 q3 h3 q3 3
On planar surface y S xy f dr S xy f x f y 2 dxdy S S xy in (xy-)plane x Stokes s Theorem S S f dr f d s S any surface ds S Example: 0 D 0 D B dr I I B d s j j d s S S S B j j 0 D differential form of Ampère s Law
Divergence Theorem z SV w ds V wdv V is any volume and S is the surface that encloses V. Surface element ds points outward. x y Example: S V 0 V 0 V 0 Qenclosed E d s E dv dv E differential form of Gauss s Law
Laplacian of /r 2 To see this: 2 r 4 ( r) f b and So we see that: r r b r b 2 2 2 2 3/ 2 2 3b 2 0 ( r 0) f b 2 2 5/ 2 2 2 r b r b ( r 0) f dv f ds 4 b V S V b b 0 b 0 r b r 2 2 lim 4 b0 2 2 r (Another approach would be to combine the Gauss s Law for inverse square central fields with the regularization of an integral by spreading out the sources of the field.)