PHY 205 Mathematical Physics

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1

Name ... lD ... .

1. Use the divergence theorem to calculate the flux of the vector field F =

<

xy, yz, xz > through the surface bounded by 4- x² -y² = z and the planes z = 3. (10 points)

2

Name ... 10 ... .

2.1

Show that F(x,y) is irrotational and incompressible. (6 points)

[(

x2 -y2

)

A 2xy A]

F(x,y)

=

A

1 - (xz+y2)2

i - (xz+y2)2l

2.2 Wii game video stick thing work with accelerometers. Accelerometer data give us

a=

^-3

cos(t) i -

³

sin(t) J + 2k

standing initially at (3, 0, O) and initial velocity is

v =

^3i.

Find the position vector

r(t).

(5 points)

3

Name ... 10 ... ..

3. The acceleration vector of a spaceship is a(t) = (2t, 0, -sin(t)) for all t

~

0 and the specific initial velocity and position are v (0) = (0, 0, 1) and r(O) = (1, 2, 300).

a) Find the velocity function

v(t)

of the spaceship (2 points)

b) Find the tangential component aT and the normal component aN of the acceleration (8 points) c) Compute the position of the space ship at time t = n/2

(4

points)

4

Name ... lD ... .

4. Use Green's theorem to calculate the line integral

p(v'l +

x³)dx

+

(2xy)dy where C is the triangle with vertices (O, O), (1, O) and (1, 3) oriented clockwise. (10 points)

5

Name ... 10 ... .

5. Calculate

f F.

dr, F(a:, Y, ,z)

=

(2:cy;,:!

+ w·'"!I,

;z:² i:i

+

xerfl, ;b·²yz:²

+

co:; .:)where and C is the arc of a helix parametrized

by

c(t)=(cost,sint,t) for O ~ t ~

n/2. (12

points)

6

Name ... I0 ... .

6. Use Stokes' Theorem to evaluate

J F. dr

,where F(x, y, z) = (e-x, e', ez) and C is the boundary of the part of the plane 2x + y + 2z = 2 in the first octant. (C is oriented counterclockwise when viewed from above.) (10 points)

7

Name ... lD ... .

Consider the periodic function f(x) defined

by

f(x)

=

^{1 - x, 0} ~ x

<

2 and f(x+2)

=

^f(x).

7.1 Sketch the graph of the function f(x)

=

1 - x on the interval -4 ~ x ~ 4. (3 points)

7.2Calculate Fourier series for the function f(x) (10 points)

8

Name ... 10 ... .

Differentf at ion Formulas:

l.-(x1=l d

dx

2.

-(ax)=

d

,l

dx

l

-,-(x")

d =

^nx"-1

dx

. d. . .

't. -(cosx)

= -smx

dx

), -d d

(sin x)

=

^{LOS X}

X

d . '

7. -kot

x)

= -csc- ,\

dx

8 d .

._. -(sec

x)

= .:..ec

^x

can

x

dx

d .

9.

-(CS(' x)

=

^-cs\:

xkot

xl

dx

,I

I

10. -(In x) = -

dx .,·

11. -(,•") J = c·

dx

12.

^-(11')

d = (In "k.

dx

I ' d ' .

^{-1 •}

. ,. -1

!Sltl X)= I ,

< X

v 1 - x·

, d

^-I

H.

-(tan x)

= - -

dx I +x

¹

d 1

15. -(sec-• x) =

-:--'.-:--'.;:::;:==

dx lxl J.\

^{2 -}

I

Integration Formulas:

l .. / ld.1 =x + C

2. /

tl

dx =

^,1.\

+ C

3 .. l x" dx = ;: ~ : + C, 11 :/= - 1

·I.

/:,;inxd.,= -cosx+(·

~ .. l

^en\.\

dx= :-in x + C

( I•

^).

~

,;J'\, \,,,

._ ::. \•

-

d \' --

""' ^l'\(1 \. .. , ^V

+ (''

• , .. /

7 . . / cs(

² x

dx = - ,:•

¹¹

x + (:

8 . . / S<.'C X(t:lll .\)

d.\ =

'.">{'( x

+ C 9 .. /

,::_sc x(,:.ot

x} dx =

^-cs..:

x + C

10.

f .!.dx=lnlxl + C

.

.\

11 .. / t''dx=

t>'

+ C

I 2. 1· ,r' dx = ~ In

_,.1

+ C a ·.- 0,

^<1

:/= I

/

, I , .

-1

c·

13.

.i(l.\'= $1()

x+ .

. ./1-x

! I i -1 c·

1,L - - ,

t

.,=

^{t,u1 x}

+ .

• l +.r

15. ;· 1

, dx = sec-

¹

x + C

. lxl ,./

^{x~ -}

I

9