Angular Velocity: Sect Overview only. For details, see text! Consider a particle moving on arbitrary path in space: –At a given instant, it can be considered as moving in a plane, circular path about an axis  Instantaneous Rotation Axis. In an infinitesimal time dt, the path can be represented as infinitesimal circular arc. As the particle moves in circular path, it has angular velocity: ω  (dθ/dt)  θ

Consider a particle moving in an instantaneously circular path of radius R. (See Fig.): –Magnitude of Particle Angular Velocity: ω  (dθ/dt)  θ –Magnitude of Linear Velocity (linear speed): v = R(dθ/dt) = Rθ = Rω

Particle moving in circular path, radius R. (Fig.): Angular Velocity: ω  θ Linear Speed: v = Rω (1) Vector direction of ω  normal to the plane of motion, in the direction of a right hand screw. (Fig.). Clearly : R = r sin(α) (2) (1) & (2)  v = rωsin(α) So (for detailed proof, see text!): v = ω  r

Gradient (Del) Operator: Sect Overview only. For details, see text! The most important vector differential operator:   grad   A Vector which has components which are differential operators.  Gradient operator. In Cartesian (rectangular) coordinates:   ∑ i e i (∂/∂x i ) (1) NOTE! (For future use!)  is much more complicated in cylindrical & spherical coordinates (see Appendix F)!!

 can operate directly on a scalar function  (  gradient of  Old Notation:  = grad  ):  = ∑ i e i (∂  /∂x i ) A VECTOR!  can operate in a scalar product with a vector A (  divergence of A; Old:  A = div A ):  A = ∑ i (∂A i /∂x i ) A SCALAR!  can operate in a vector product with a vector A (  curl of A; Old:   A = curl A ): (  A) i = ∑ j,k ε ijk (∂A k /∂x j ) A VECTOR! ( Older:   A = rot A) Obviously, A = A(x,y,z)

Physical interpretation of the gradient  : (Fig) The text shows that  has the properties: 1. It is  surfaces of constant  2. It is in the direction of max change in  3. The directional derivative of  for any direction n is n  = (∂  /∂n)  (x,y)   Contour plot of  (x,y)

The Laplacian Operator The Laplacian is the dot product of  with itself:  2   ;  2  ∑ i (∂ 2 /∂x i 2 ) A SCALAR! The Laplacian of a scalar function   2   ∑ i (∂ 2  /∂x i 2 )

Integration of Vectors: Sect Overview only. For details, see text! Types of integrals of vector functions: A = A(x,y,z) = A(x 1,x 2,x 3 ) = (A 1,A 2,A 3 ) Volume Integral (volume V, differential volume element dv) (Fig.): ∫ V A dv  ( ∫ V A 1 dv, ∫ V A 2 dv, ∫ V A 3 dv)

Surface Integral (surface S, differential surface element da) (Fig.) ∫ S A  n da, n  Normal to surface S

Line integral (path in space, differential path element ds) (Fig.): ∫ BC A  ds  ∫ BC ∑ i A i dx i

Gauss’s Theorem or Divergence Theorem (for a closed surface S surrounding a volume V) See figure; n  Normal to surface S ∫ S A  n da = ∫ V  A dv Physical Interpretation of  A The net “amount” of A “flowing” in & out of closed surface S

Stoke’s Theorem (for a closed loop C surrounding a surface S) See Figure; n  Normal to surface S ∫ C A  ds = ∫ S (  A)  n da Physical Interpretation of  A The net “amount” of “rotation” of A