= ( new notation ) A. n A ds.

Transcription

1 .6. Vector Fields (continued).6.2. The divergence of a vector field. Recall div A = A + A 2 + A 3 = ( new notation ) A. 3 Recall Gauss Theorem: div A dv = A n ds. V Dividing the equation by the volume V and taking the limit V 0sothatV shrinks toapoint, we find A() = lim V 0 V V V n A ds. So we have found a coordinate-independent representation of the divergence. Using the definition of flu, we see that the integral over V is the total flu through the surface V. This total flu over volume V is the flu per unit volume. In the limit V 0, the limiting value measures the flu production per unit volume at the point. This is the real meaning of the divergence. If it is positive, then it is called a source. If it is negative, then it is called a sink. If it is zero in a domain, then there is no source or sink, and it is called divergence free. Eample.6.2a. Let A(r) =q r r 3 where r denotes the norm of r. Itisaneercisethat div A =0 (r 0) at every point ecept r = 0. We are interested to find the flu through a sphere S centered at the origin. We know that the unit eterior normal to a sphere is given by n = r r. So we have A n ds S = q r r ds S r 3 r = q ds S r 2 =4πq. ()

2 If q>0, it is a source (fountain). If q<0, it is a sink. By Gauss Theorem, we can see that the flu through any surface is 4πq if the surface encloses the origin. The flu is zero if the surface does not enclose the origin. We also see that the flu is the same no matter how small the surface is as long as it contains the origin. This vector field is smooth every where away from the origin, and the origin is a point source/sink. See Figure.6.2. Source Sink Figure.6.2. Flu and source/sink The curl of a vector field. Recall we have introduced the curl of a vector in association with Stokes Theorem: curl A =( A 3 = 2 A 2 3, A 3 A 3, A 2 A 2 ) i i 2 i A A 2 A 3 We state without proof that the curl has a coordinate-independent representation: curl A = lim n A ds (= A). V 0 V V You can see a lot by looking Jogi Berra. Can you see a common theme in the three formulas for φ, A, A? Once you treat =(, 2, 3 )asa vector, many formulas involving first derivatives are a lot easier to memorize. For eample, once you know the determinant formula for A B, you can replace A by to find B. 2

3 We see the real meaning of the curl in the net eample. Eample.6.3a. Consider a rigid body rotating about a fied point O with angular velocity w. See Figure.6.3. The velocity of a point with position vector r is given by v = w r. w r v Figure.6.3. Curl is twice angular velocity. Let us calculate the curl of v. curl v = 2 v 3 3 v 2 (w is independent of r), and similarly (homework) = 2 (w 2 w 2 ) 3 (w 3 w 3 ) (2) =2w curl 2 v =2w 2 curl 3 v =2w 3. (3) It follows that curl v =2w. That is, the curl of the velocity field of a rotating body equals twice the angular velocity of the body..7. Coordinate transformations. We deal with coordinate transformations between rectangular coordinate systems, which play an important role in the definition of tensors. Preliminary remark on tensors. Tensors are physical quantities that eist independent of coordinate systems. Scalar quantities are called zeroth-order tensors (e.g., temperature); vectors are called first-order tensors (e.g., velocity); secondorder tensors can all be represented by 3 3 matrices (e.g., the stress tensor), but 3

4 not all 3 3 matrices are tensors. True telephone numbers versus a string of digits (82) provides a metaphor. (That is, not every such a string of digits is a telephone number.) Tensors of orders greater than 2 cannot be represented by 3 3 matrices. An n-th-order tensor requires 3 n real numbers and is invariant under change of coordinate systems. The requirement of the invariance is natural since physical observables are invariant under change of coordinate systems. Suppose we have two orthonormal bases: i, i 2, i 3, and i, i 2, i 3, and two origin O and O to form two rectangular coordinate systems K and K.Let apointm inspacehavetherepresentation r = i + 2 i i 3 r = i + 2 i i 3. (4) Note the equations: r = r + r 0, r 0 = OO r = r + r 0, r 0 = O O, (5) where the vector r 0 is a vector from O to O and r 0 = r 0. See Figure.7.. (Figure.7..) 4

5 M 3 3 r i 3 r K 2 K i 3 O i i 2 r 0 i O 2 i 2 Figure.7.. Coordinate transformation. Now we use the summation convention: a repeated inde in a product is automatically summed: 3 k i k = k i k. k= Thus, the equations in (5) can be written k i k k i k = k i k + 0k i k = k i k + 0k i k (6) where 0k are the coordinates of r 0 with respect to the old system K, etc. Take inner product with i l or i l in equations (6) and note the Kronecker delta function 0, k l, i k i l = δ kl = (7), k = l. We find l = k (i k i l)+ 0l l = k (i k i l )+ 0l. (8) 5

6 We introduce new notations i k i l = cos(i k, i l )=α k l. (9) Thus i k i l = i l i k = α l k. (0) Therefore l = α k l k + 0l l = α l k k + 0l. () The first equation of () is the transformation from K to K, while the second equation of () is the inverse transformation from K to K. Note the inde summed in the second equation is the second inde of α, while the inde summed in the first equation of () is the first inde. If you want to know the parallel notation in matri form, then the equation l = α l k k + 0l can be written as =(α l k) + 0 where all,, 0 are written in column vector form, and (α l k) is written in matri form. -End of Lecture 6 6