7/25/2019 assignment_02_081007

1/1

HOMEWORK ASSIGNMENT: Friday, August 10, 2007

Problem 1:We have the equations of motion, (the conservations of mass & linear mo-mentum), for an inviscid fluid

t

+vk;k+vk;k = 0, (1)

t

vi +vkvi;k

=

P gki;k

. (2)

Here, t denotes time, = (um, t) the mass density, P = P(um, t) the pres-sure, and vi = vi(um, t) the velocity of the fluid. (We will derive theseequations later in the class.)

(a) Write the equations for and for v1 (the first component equation)in the spherical coordinate sytem whose covariant/natural base vectors aregiven below.

(b) Write further the equations obtained in (a) in terms of the physicalcomponents of the velocity. (The notion of physical components is explainedin this weeks tutoral sheet.)

In a spherical coordinate system with (r= u1, = u2, = u3), the covari-ant base vectors are

g1= sin(u2)

cos(u3)i1+ sin(u

3)i2

+ cos(u2)i3,

g2= u1cos(u2)

cos(u3)i1+ sin(u

3)i2 u1sin(u2)i3,

g3= u1sin(u2)

sin(u3)i1+ cos(u

3)i2

. (3)

(If you feel uncomfortable about the index taking values 1, 2, 3 in the sphericalcoordinate system, you can try an alternative, like using r, , as the valuesfor the indexes i, j, k..., while keepping i1, i2, i3 fixed/intact.)

Problem 2:The problems assigned in the class. That is, verify Equations (15) and (17)in the lecture note of class 04 handout.pdf. (You can select several toverify, but at least two from (17) one about transformation and one aboutlowering/raising the indexes.)

Problem 3: (Optional)Try to write Akj

kl;m. (From the patterns of the differentiation of the first

order contravariant and the first order covariant tensors.)

1